Clep college algebra
Simplify:(-10x-100)+(-8x+8)-(-4x+1)
The Correct Answer: -14x-93+(-8x+8) is the same as +(1)(-8x+8) therefore you can just remove the parentheses and write -8x+8. -(-4x+1) is the same as +(-1)(-4x+1) so we must use distribution to obtain +4x-1. Therefore the entire expression can be re-written,-10x-100-8x+8+4x-1 Next combine like terms:-10x-8x+4x = -14x-100+8-1 = -93Therefore the final answer is -14x-93
Simplify:(-8x-80)-(9x-5)
The Correct Answer: -17x-75given:(-8x-80)-(9x-5) Recall that -(9x-5) is the same as +(-1)(9x-5).Therefore, you must distribute -1 into the parentheses and get for the entire expression:-8x-80-9x+5Now combine like terms as follows-8x-80-9x+5 Then we get the following solution:-17x-75
Find 6+3x2 when x=-8
The Correct Answer: 198 Take the given value of x, which is -8,and plug it into the expression 6+3x2 like this: 6+3(-8)2 = 198
Find 3x2-7x-6 when x=-2
The Correct Answer: 20Since x=-2, replace every x value with -2 in the expression, 3x2-7x-6.Like this:3(-2)2-7(-2)-6 = 20
Find 8+3x2 when x=9
The Correct Answer: 251 Since x=9, find 8+3x2 by replacing every x value with 9, like this:8+3(9)2 = 251
Find -8x2-2x+6 when x=0
The Correct Answer: 6 Take the given x value of 0 and plug in 0 for every x you see in the statement -8x2-2x+6 Like this: -8(0)2-2(0)+6 this gives you an answer of 6
Factor:11x2-4x-7
Correct Answer: (11x+7)(x-1)Given:11x2-4x-7This is the same as:11x2-4x-7 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ )(Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses. ( __ ? __ )( __ ? __ ) How?: Look at the last sign: 11x2-4x-7 if the last sign is (-), as it is here, then the signs alternate: ( __ - __ )( __ + __ ) NOTICE:If it is (+), the two signs will be the same. Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ). (Step 2) Next determine the first and last numbers for each parentheses. ( ? - ? )( ? + ? ) How? List the factors of the first and last terms: 11x2-4x-7 Factors of 11x2 Factors of 7 Multiply Across x 7 7x 11x 1 11x Since the last sign, as we determined above, is (-), the signs are different in either parentheses and therefore the factors in the above table must SUBTRACT to the middle term of 4x. And since 11x - 7x = 4x, we have picked the correct factors of the first and last terms. The factors of the first and last terms are arranged in the parentheses in this way every time: ( # +/- # )( # +/- # ) Now look to see if there is a (+) or (-) sign in the middle: 11x2-4x-7 The middle term is (-), and since the A term is 11, we need to perform a "check" before we know where to place the (-) sign. If we place the (-) sign in the first parentheses: (11x-7)(x+1) Then Check: (1) Multiply the inner terms: -7 * x = -7x (2) Multiply the outer terms: 11x * (+1) = 11x Then add -7x+11x = +4x * Therefore this is wrong; our middle term is NOT (+) If we place the (-) sign in the second parentheses: (11x+7)(x-1) Then Check: (1)Multiply the inner terms:+7 * (x) = +7x (2) Multiply the outer terms:11x * (-1) = -11x Then add +7x-11x = -4x * This IS our middle term, therefore this is correct!Therefore our solution is:(11x+7)(x-1), which can also be written (x-1)(11x+7).
Factor:2x2+13x+15
Correct Answer: (2x+3)(x+5)Given:2x2+13x+15This is the same as:2x2+13x+15 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ ) (Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses. ( __ ? __ )( __ ? __ ) How?: Look at the last sign: 2x2+13x+15 If it is (+) as in our case, the two signs will be the same. Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ). Next look at the middle sign: If the middle sign is (+) as in our case, the signs are both (+). ( __ + __ )( __ + __ ). NOTICE: if the last sign was (-) then the signs would alternate regardless of the middle sign, like this: ( __ - __ )( __ + __ ) (Step 2) Next determine the first and last numbers for each parentheses. ( ? + ? )( ? + ? ) How? List the factors of the first and last terms: 2x2+13x+15 Factors of 2x2 Factors of 15 Multiply Across 2x 5 10x x 3 3x Since the last sign, as we determined above, is (+), the signs are the same in either parentheses and therefore the factors in the above table must ADD to the middle term of 13x. And since 10x + 3x = 13x, we have picked the correct factors of the first and last terms. The factors of the first and last terms are arranged in the parentheses in this way every time: ( # +/- # )( # +/- # )Therefore our solution is:(2x+3)(x+5), which can also be written (x+5)(2x+3).
Factor:4x2-1
Correct Answer: (2x-1)(2x+1)Given:4x2-1 This is the same as:(2x)2-(1)2 ⇒ Since 22x2-12 = 4x2-1 Which is of the form:a2-b2 When you see a polynomial that can be arranged in this way, it can be solved using The Difference of Squares Formula: a2 - b2 = (a-b)(a+b) (1) First identify your a and b terms. (2x)2-(1)2Therefore a = 2x and b = 1 (a-b)(a+b) (2) Then plug these values into the difference of squares formula.Hence the factored form of 4x2-1 is:(2x-1)(2x+1) which can also be correctly written as (2x+1)(2x-1). WARNING! A polynomial of the form 4x2+1 is NOT factor-able since a "sum of squares" formula a2+b2 DOES NOT EXIST!
Factor:2x2-11x+5
Correct Answer: (2x-1)(x-5)Given:2x2-11x+5This is the same as:2x2-11x+5 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ )(Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses. ( __ ? __ )( __ ? __ ) How?: Look at the last sign: 2x2-11x+5 If it is (+) as in our case, the two signs will be the same. Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ). Next look at the middle sign: If the middle sign is (-) as in our case, the signs are both (-). ( __ - __ )( __ - __ ). NOTICE: if the last sign was (-) then the signs would alternate regardless of the middle sign, like this: ( __ - __ )( __ + __ ) (Step 2) Next determine the first and last numbers for each parentheses. ( ? - ? )( ? - ? ) How? List the factors of the first and last terms: 2x2-11x+5 Factors of 2x2 Factors of 5 Multiply Across 2x 5 10x x 1 1x Since the last sign, as we determined above, is (+), the signs are the same in either parentheses and therefore the factors in the above table must ADD to the middle term of 11x. And since 10x + 1x = 11x, we have picked the correct factors of the first and last terms. The factors of the first and last terms are arranged in the parentheses in this way every time: ( # +/- # )( # +/- # ) Therefore our solution is:(2x-1)(x-5), which can also be written (x-5)(2x-1).
Factor:3x2+2x-1
Correct Answer: (3x-1)(x+1)Given:3x2+2x-1This is the same as:3x2+2x-1 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ )(Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses. ( __ ? __ )( __ ? __ ) How?: Look at the last sign: 3x2+2x-1 if the last sign is (-), as it is here, then the signs alternate: ( __ - __ )( __ + __ ) NOTICE:If it is (+), the two signs will be the same. Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ). (Step 2) Next determine the first and last numbers for each parentheses. ( ? - ? )( ? + ? ) How? List the factors of the first and last terms: 3x2+2x-1 Factors of 3x2 Factors of 1 Multiply Across 3x 1 3x x 1 1x Since the last sign, as we determined above, is (-), the signs are different in either parentheses and therefore the factors in the above table must SUBTRACT to the middle term of 2x. And since 3x - 1x = 2x, we have picked the correct factors of the first and last terms. The factors of the first and last terms are arranged in the parentheses in this way every time: ( # +/- # )( # +/- # ) Now look to see if there is a (+) or (-) sign in the middle: 3x2+2x-1The middle term is (+), and since the A term is 3 (and NOT 1), we need to perform a "check" before we know where to place the (-) sign. If we place the (-) sign in the first parentheses: (3x-1)(x+1) Then Check: (1) Multiply the inner terms: -1 * x = -1x (2) Multiply the outer terms: 3x * (+1) = +3x Then add -1x+3x = +2x * * This IS our middle term, therefore this is correct! Therefore our solution is:(3x-1)(x+1), which can also be written (x+1)(3x-1).
Factor:16x2-49
Correct Answer: (4x-7)(4x+7)Given:16x2-49 This is the same as:(4x)2-(7)2 ⇒ Since 42x2-72 = 16x2-49 Which is of the form:a2-b2 When you see a polynomial that can be arranged in this way, it can be solved using The Difference of Squares Formula: a2 - b2 = (a-b)(a+b) (1) First identify your a and b terms. (4x)2-(7)2Therefore a = 4x and b = 7 (a-b)(a+b) (2) Then plug these values into the difference of squares formula.Hence the factored form of 16x2-49 is:(4x-7)(4x+7) which can also be correctly written as (4x+7)(4x-7). WARNING! A polynomial of the form 16x2+49 is NOT factor-able since a "sum of squares" formula a2+b2 DOES NOT EXIST!
Factor:5x2+16x+11
Correct Answer: (5x+11)(x+1)Given:5x2+16x+11This is the same as:5x2+16x+11 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ ) (Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses. ( __ ? __ )( __ ? __ ) How?: Look at the last sign: 5x2+16x+11 If it is (+) as in our case, the two signs will be the same. Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ). Next look at the middle sign: If the middle sign is (+) as in our case, the signs are both (+). ( __ + __ )( __ + __ ). NOTICE: if the last sign was (-) then the signs would alternate regardless of the middle sign, like this: ( __ - __ )( __ + __ ) (Step 2) Next determine the first and last numbers for each parentheses. ( ? + ? )( ? + ? ) How? List the factors of the first and last terms: 5x2+16x+11 Factors of 5x2 Factors of 11 Multiply Across 5x 1 5x x 11 11x Since the last sign, as we determined above, is (+), the signs are the same in either parentheses and therefore the factors in the above table must ADD to the middle term of 16x. And since 11x + 5x = 16x, we have picked the correct factors of the first and last terms. The factors of the first and last terms are arranged in the parentheses in this way every time: ( # +/- # )( # +/- # ) Therefore our solution is:(5x+11)(x+1), which can also be written (x+1)(5x+11).
Factor:5x2-39x+28
Correct Answer: (5x-4)(x-7)Given:5x2-39x+28This is the same as:5x2-39x+28 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ )(Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses. ( __ ? __ )( __ ? __ ) How?: Look at the last sign: 5x2-39x+28 If it is (+) as in our case, the two signs will be the same. Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ). Next look at the middle sign: If the middle sign is (-) as in our case, the signs are both (-). ( __ - __ )( __ - __ ). NOTICE: if the last sign was (-) then the signs would alternate regardless of the middle sign, like this: ( __ - __ )( __ + __ )(Step 2) Next determine the first and last numbers for each parentheses. ( ? - ? )( ? - ? ) How? List the factors of the first and last terms: 5x2-39x+28 Factors of 5x2 Factors of 28 Multiply Across 5x 7 35x x 4 4x Since the last sign, as we determined above, is (+), the signs are the same in either parentheses and therefore the factors in the above table must ADD to the middle term of 39x. And since 35x + 4x = 39x, we have picked the correct factors of the first and last terms. The factors of the first and last terms are arranged in the parentheses in this way every time: ( # +/- # )( # +/- # ) Therefore our solution is:(5x-4)(x-7), which can also be written (x-7)(5x-4).
Factor:30x2+53x+8
Correct Answer: (6x+1)(5x+8)Given:30x2+53x+8This is the same as:30x2+53x+8 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ )(Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses. ( __ ? __ )( __ ? __ ) How?: Look at the last sign: 30x2+53x+8 If it is (+) as in our case, the two signs will be the same. Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ). Next look at the middle sign: If the middle sign is (+) as in our case, the signs are both (+). ( __ + __ )( __ + __ ). NOTICE: if the last sign was (-) then the signs would alternate regardless of the middle sign, like this: ( __ - __ )( __ + __ )(Step 2) Next determine the first and last numbers for each parentheses. ( ? + ? )( ? + ? ) How? List the factors of the first and last terms: 30x2+53x+8 Factors of 30x2 Factors of 8 Multiply Across 6x 8 48x 5x 1 5x Since the last sign, as we determined above, is (+), the signs are the same in either parentheses and therefore the factors in the above table must ADD to the middle term of 53x. And since 48x + 5x = 53x, we have picked the correct factors of the first and last terms. The factors of the first and last terms are arranged in the parentheses in this way every time: ( # +/- # )( # +/- # )Therefore our solution is:(6x+1)(5x+8), which can also be written (5x+8)(6x+1).
Factor:6x2+23x+17
Correct Answer: (6x+17)(x+1)Given:6x2+23x+17This is the same as:6x2+23x+17 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ )(Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses. ( __ ? __ )( __ ? __ )How?: Look at the last sign: 6x2+23x+17 If it is (+) as in our case, the two signs will be the same. Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ). Next look at the middle sign: If the middle sign is (+) as in our case, the signs are both (+). ( __ + __ )( __ + __ ). NOTICE: if the last sign was (-) then the signs would alternate regardless of the middle sign, like this: ( __ - __ )( __ + __ ) (Step 2) Next determine the first and last numbers for each parentheses. ( ? + ? )( ? + ? ) How? List the factors of the first and last terms: 6x2+23x+17 Factors of 6x2 Factors of 17 Multiply Across 6x 1 6x x 17 17x Since the last sign, as we determined above, is (+), the signs are the same in either parentheses and therefore the factors in the above table must ADD to the middle term of 23x. And since 17x + 6x = 23x, we have picked the correct factors of the first and last terms. The factors of the first and last terms are arranged in the parentheses in this way every time: ( # +/- # )( # +/- # ) Therefore our solution is:(6x+17)(x+1), which can also be written (x+1)(6x+17).
Factor:12xy+2by+6ax+ab
Correct Answer: (6x+b)(2y+a)Given:12xy+2by+6ax+ab When factoring a polynomial consisting of 4 terms, Factor by Grouping: (Step 1) Determine if the first two terms have a Greatest Common Factor between them, and that the last two terms have a Greatest Common Factor between them. 12xy+2by+6ax+ab The first two terms: 12xy + 2by have 2y in common and the last two terms: 6ax + ab have a in common Note: These terms can be rearranged by simply moving the terms around in order to get the first two and last two terms to have a Greatest Common Factor. (Step 2) Then group the first two terms and the last two terms. (12xy+2by)+(6ax+ab) Make sure you have a + between each group.(Step 3) Factor out the Greatest Common Factor from the first two and then the last two terms. 2y(6x+b)+a(6x+b) *The ideal situation is having the terms in each parentheses matchso that we can factor them out of each "group" Therefore the answer is: (6x+b)(2y+a)
Factor:49xy+56x-7y-8
Correct Answer: (7x-1)(7y+8)Given:49xy+56x-7y-8 When factoring a polynomial consisting of 4 terms, Factor by Grouping: (Step 1) Determine if the first two terms have a Greatest Common Factor between them, and that the last two terms have a Greatest Common Factor between them. 49xy+56x+-7y-8 The first two terms: 49xy + 56x have 7x in common and the last two terms: -7y - 8 do not have a Greatest Common Factor in common Note: These terms can be rearranged by simply moving the terms around in order to get the first two and last two terms to have a Greatest Common Factor. New arrangement:Since 49xy - 7y have a GCF of 7y we list these terms firstThen since 56x-8 have a GCF of 8 we list these terms secondTherefore we have: 49xy-7y+56x-8 (Step 2) Then group the first two terms and the last two terms. (49xy - 7y)+(56x-8) Make sure you have a + between each group. (Step 3) Factor out the Greatest Common Factor from the first two and then the last two terms. 7y(7x-1)+8(7x-1) *The ideal situation is having the terms in each parentheses matchso that we can factor them out of each "group" Therefore the answer is: (7x-1)(7y+8)
Factor:7x2+13x-2
Correct Answer: (7x-1)(x+2)Given:7x2+13x-2This is the same as:7x2+13x-2 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ ) (Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses. ( __ ? __ )( __ ? __ ) How?: Look at the last sign: 7x2+13x-2 if the last sign is (-), as it is here, then the signs alternate: ( __ - __ )( __ + __ ) NOTICE:If it is (+), the two signs will be the same. Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ). (Step 2) Next determine the first and last numbers for each parentheses. ( ? - ? )( ? + ? ) How? List the factors of the first and last terms: 7x2+13x-2 Factors of 7x2 Factors of 2 Multiply Across 7x 2 14x x 1 1x Since the last sign, as we determined above, is (-), the signs are different in either parentheses and therefore the factors in the above table must SUBTRACT to the middle term of 13x. And since 14x - 1x = 13x, we have picked the correct factors of the first and last terms. The factors of the first and last terms are arranged in the parentheses in this way every time: ( # +/- # )( # +/- # ) Now look to see if there is a (+) or (-) sign in the middle: 7x2+13x-2 The middle term is (+), and since the A term is 7 (and NOT 1), we need to perform a "check" before we know where to place the (-) sign. If we place the (-) sign in the first parentheses: (7x-1)(x+2) Then Check: (1) Multiply the inner terms: -1 * x = -1x (2) Multiply the outer terms: 7x * (+2) = +14x Then add -1x+14x = +13x * * This IS our middle term, therefore this is correct! Therefore our solution is:(7x-1)(x+2), which can also be written (x+2)(7x-1).
Factor:8x2-23x-3
Correct Answer: (8x+1)(x-3)Given:8x2-23x-3This is the same as:8x2-23x-3 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ ) (Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses. ( __ ? __ )( __ ? __ ) How?: Look at the last sign: 8x2-23x-3 if the last sign is (-), as it is here, then the signs alternate: ( __ - __ )( __ + __ ) NOTICE:If it is (+), the two signs will be the same. Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ). (Step 2) Next determine the first and last numbers for each parentheses. ( ? - ? )( ? + ? ) How? List the factors of the first and last terms: 8x2-23x-3 Factors of 8x2 Factors of 3 Multiply Across 8x 3 24x x 1 1x Since the last sign, as we determined above, is (-), the signs are different in either parentheses and therefore the factors in the above table must SUBTRACT to the middle term of 23x. And since 24x - x = 23x, we have picked the correct factors of the first and last terms. The factors of the first and last terms are arranged in the parentheses in this way every time: ( # +/- # )( # +/- # ) Now look to see if there is a (+) or (-) sign in the middle: 8x2-23x-3 The middle term is (-), and since the A term is 8, we need to perform a "check" before we know where to place the (-) sign. If we place the (-) sign in the first parentheses: (8x-1)(x+3) Then Check: (1) Multiply the inner terms: -1 * x = -1x (2) Multiply the outer terms: 8x * (+3) = 24x Then add -x+24x = +4x * Therefore this is wrong; our middle term is NOT (+) If we place the (-) sign in the second parentheses: (8x+1)(x-3) Then Check: (1)Multiply the inner terms:+1 * (x) = +x (2) Multiply the outer terms:8x * (-3) = -24x Then add -24x+x = -23x * This IS our middle term, therefore this is correct! Therefore our solution is:(8x+1)(x-3), which can also be written (x-3)(8x+1).
Factor:81xy+45by+63ax+35ab
Correct Answer: (9y+7a)(9x+5b)Given:81xy+45by+63ax+35ab When factoring a polynomial consisting of 4 terms, Factor by Grouping: (Step 1) Determine if the first two terms have a Greatest Common Factor between them, and that the last two terms have a Greatest Common Factor between them. 81xy+45by+63ax+35ab The first two terms: 81xy + 45by have 9y in common and the last two terms: 63ax + 35ab have 7a in common Note: These terms can be rearranged by simply moving the terms around in order to get the first two and last two terms to have a Greatest Common Factor. (Step 2) Then group the first two terms and the last two terms. (81xy+45by)+(63ax+35ab) Make sure you have a + between each group. (Step 3) Factor out the Greatest Common Factor from the first two and then the last two terms. 9y(9x+5b)+7a(9x+5b) *The ideal situation is having the terms in each parentheses matchso that we can factor them out of each "group" Therefore the answer is: (9x+5b)(9y+7a) This is the same as: (9y+7a)(9x+5b)
Factor:x2-5x-6
Correct Answer: (x+1)(x-6)Given:x2-5x-6This is the same as:1x2-5x-6 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ ) (Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses. ( __ ? __ )( __ ? __ ) How?: Look at the last sign: x2-5x-6 if the last sign is (-), as it is here, then the signs alternate: ( __ - __ )( __ + __ ) NOTICE:If it is (+), the two signs will be the same. Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ). (Step 2) Next determine the first and last numbers for each parentheses. ( ? - ? )( ? + ? ) How? List the factors of the first and last terms: x2-5x-6 Factors of x2 Factors of 6 Multiply Across x 6 6x x 1 1x Since the last sign, as we determined above, is (-), the signs are different in either parentheses and therefore the factors in the above table must SUBTRACT to the middle term of 5x. And since 6x - 1x = 5x, we have picked the correct factors of the first and last terms. The factors of the first and last terms are arranged in the parentheses in this way every time: ( # +/- # )( # +/- # ) Now look to see if there is a (+) or (-) sign in the middle: x2-5x-6 Because the middle term is (-), let the larger factor be in the parentheses with the (-) sign. Therefore our solution is:(x+1)(x-6), which can also be written (x-6)(x+1).
Factor:x2-36
Correct Answer: (x+6)(x-6)Given:x2-36 This is the same as:(x)2-(6)2 ⇒ Since x2-62 = x2-36 Which is of the form:a2-b2 When you see a polynomial that can be arranged in this way, it can be solved using The Difference of Squares Formula: a2 - b2 = (a-b)(a+b) (1) First identify your a and b terms. (x)2-(6)2Therefore a = x and b = 6 (a-b)(a+b) (2) Then plug these values into the difference of squares formula.Hence the factored form of x2-36 is:(x+6)(x-6) which can also be correctly written as (x-6)(x+6). WARNING! A polynomial of the form x2+36 is NOT factor-able since a "sum of squares" formula a2+b2 DOES NOT EXIST!
Factor:x2+4x-12
Correct Answer: (x-2)(x+6)Given:x2+4x-12This is the same as:1x2+4x-12 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ )(Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses. ( __ ? __ )( __ ? __ ) How?: Look at the last sign: x2+4x-12 if the last sign is (-), as it is here, then the signs alternate: ( __ - __ )( __ + __ ) NOTICE:If it is (+), the two signs will be the same. Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ). (Step 2) Next determine the first and last numbers for each parentheses. ( ? - ? )( ? + ? ) How? List the factors of the first and last terms: x2+4x-12 Factors of x2 Factors of 12 Multiply Across x 6 6x x 2 2x Since the last sign, as we determined above, is (-), the signs are different in either parentheses and therefore the factors in the above table must SUBTRACT to the middle term of 4x. And since 6x - 2x = 4x, we have picked the correct factors of the first and last terms. The factors of the first and last terms are arranged in the parentheses in this way every time: ( # +/- # )( # +/- # ) Now look to see if there is a (+) or (-) sign in the middle: x2+4x-12 Because the middle term is (+), let the larger factor be in the parentheses with the (+) sign.Therefore our solution is:(x-2)(x+6), which can also be written (x+6)(x-2).
Factor:x2-8x+12
Correct Answer: (x-2)(x-6)Given:x2-8x+12This is the same as:1x2-8x+12 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ ) (Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses. ( __ ? __ )( __ ? __ ) How?: Look at the last sign: x2-8x+12 If it is (+) as in our case, the two signs will be the same. Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ). Next look at the middle sign: If the middle sign is (-) as in our case, the signs are both (-). ( __ - __ )( __ - __ ). NOTICE: if the last sign was (-) then the signs would alternate regardless of the middle sign, like this: ( __ - __ )( __ + __ ) (Step 2) Next determine the first and last numbers for each parentheses. ( ? - ? )( ? - ? ) How? List the factors of the first and last terms: x2-8x+12 Factors of x2 Factors of 12 Multiply Across x 6 6x x 2 2x Since the last sign, as we determined above, is (+), the signs are the same in either parentheses and therefore the factors in the above table must ADD up to the middle term of 8x. And since 6x + 2x = 8x, we have picked the correct factors of the first and last terms. The factors of the first and last terms are arranged in the parentheses in this way every time: ( # +/- # )( # +/- # ) Therefore our solution is:(x-2)(x-6), which can also be written (x-6)(x-2).
Factor:x2-25x+100
Correct Answer: (x-20)(x-5)Given:x2-25x+100This is the same as:1x2-25x+100 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ ) (Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses. ( __ ? __ )( __ ? __ ) How?: Look at the last sign: x2-25x+100 If it is (+) as in our case, the two signs will be the same. Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ). Next look at the middle sign: If the middle sign is (-) as in our case, the signs are both (-). ( __ - __ )( __ - __ ). NOTICE: if the last sign was (-) then the signs would alternate regardless of the middle sign, like this: ( __ - __ )( __ + __ ) (Step 2) Next determine the first and last numbers for each parentheses. ( ? - ? )( ? - ? ) How? List the factors of the first and last terms: x2-25x+100 Factors of x2 Factors of 100 Multiply Across x 20 20x x 5 5x Since the last sign, as we determined above, is (+), the signs are the same in either parentheses and therefore the factors in the above table must ADD up to the middle term of 25x. And since 20x + 5x = 25x, we have picked the correct factors of the first and last terms. The factors of the first and last terms are arranged in the parentheses in this way every time: ( # +/- # )( # +/- # ) Therefore our solution is:(x-20)(x-5), which can also be written (x-5)(x-20).
Factor:x2-4x+3
Correct Answer: (x-3)(x-1)Given:x2-4x+3This is the same as:1x2-4x+3 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ )(Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses. ( __ ? __ )( __ ? __ ) How?: Look at the last sign: 1x2-4x+3 If it is (+) as in our case, the two signs will be the same. Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ). Next look at the middle sign: If the middle sign is (-) as in our case, the signs are both (-). ( __ - __ )( __ - __ ). NOTICE:if the last sign was (-) then the signs would alternate regardless of the middle sign, like this: ( __ - __ )( __ + __ ) (Step 2) Next determine the first and last numbers for each parentheses. ( ? - ? )( ? - ? ) How? List the factors of the first and last terms: x2-4x+3 Factors of x2 Factors of 3 Multiply Across x 1 1x x 3 3x Since the last sign, as we determined above, is (+), the signs are the same in either parentheses and therefore the factors in the above table must ADD to the middle term of 4x. And since x + 3x = 4x, we have picked the correct factors of the first and last terms. The factors of the first and last terms are arranged in the parentheses in this way every time: ( # +/- # )( # +/- # )Therefore our solution is:(x-3)(x-1), which can also be written (x-1)(x-3).
Factor:6xy-18by-5ax+15ab
Correct Answer: (x-3b)(6y-5a)Given:6xy-18by-5ax+15ab When factoring a polynomial consisting of 4 terms, Factor by Grouping: (Step 1) Determine if the first two terms have a Greatest Common Factor between them, and that the last two terms have a Greatest Common Factor between them. 6xy-18by+-5ax+15ab The first two terms: 6xy - 18by have 6y in common and the last two terms: -5ax + 15ab have 5a in common Note: These terms can be rearranged by simply moving the terms around in order to get the first two and last two terms to have a Greatest Common Factor. (Step 2) Then group the first two terms and the last two terms. (6xy-18by)+(-5ax+15ab) Make sure you have a + between each group.(Step 3) Factor out the Greatest Common Factor from the first two and then the last two terms. 6y(x-3b)+5a(-x+3b) *The ideal situation is having the terms in each parentheses matchso that we can factor them out of each "group" ⇒ Therefore since (x-3b) and (-x+3b) are the same but opposite in sign, factor a -1 from the second group to make them identical. 6y(x-3b)-5a(x-3b) Now factor (x-3b) out of each group. Therefore the answer is: (x-3b)(6y-5a)
Factor:x2+x-56
Correct Answer: (x-7)(x+8)Given:x2+x-56This is the same as:1x2+x-56 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ ) (Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses. ( __ ? __ )( __ ? __ ) How?: Look at the last sign: x2+x-56 if the last sign is (-), as it is here, then the signs alternate: ( __ - __ )( __ + __ ) NOTICE:If it is (+), the two signs will be the same. Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ). (Step 2) Next determine the first and last numbers for each parentheses. ( ? - ? )( ? + ? ) How? List the factors of the first and last terms: x2+x-56 Factors of x2 Factors of 56 Multiply Across x 8 8x x 7 7x Since the last sign, as we determined above, is (-), the signs are different in either parentheses and therefore the factors in the above table must SUBTRACT to the middle term of x. And since 8x - 7x = x, we have picked the correct factors of the first and last terms. The factors of the first and last terms are arranged in the parentheses in this way every time: ( # +/- # )( # +/- # ) Now look to see if there is a (+) or (-) sign in the middle: x2+x-56 Because the middle term is (+), let the larger factor be in the parentheses with the (+) sign. Therefore our solution is:(x-7)(x+8), which can also be written (x+8)(x-7).
Factor:x2-14x+49
Correct Answer: (x-7)(x-7) which can also be written as (x-7)2Given:x2-14x+49This is the same as:1x2-14x+49 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ ) (Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses. ( __ ? __ )( __ ? __ ) How?: Look at the last sign: x2-14x+49 If it is (+) as in our case, the two signs will be the same. Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ). Next look at the middle sign: If the middle sign is (-) as in our case, the signs are both (-). ( __ - __ )( __ - __ ). NOTICE: if the last sign was (-) then the signs would alternate regardless of the middle sign, like this: ( __ - __ )( __ + __ ) (Step 2) Next determine the first and last numbers for each parentheses. ( ? - ? )( ? - ? ) How? List the factors of the first and last terms: x2-14x+49 Factors of x2 Factors of 49 Multiply Across x 7 7x x 7 7x Since the last sign, as we determined above, is (+), the signs are the same in either parentheses and therefore the factors in the above table must ADD up to the middle term of 14x. And since 7x + 7x = 14x, we have picked the correct factors of the first and last terms. The factors of the first and last terms are arranged in the parentheses in this way every time: ( # +/- # )( # +/- # ) Therefore our solution is:(x-7)(x-7), which can also be written (x-7)2).
Factor:64x2-16
Correct Answer: 16(2x-1)(2x+1) Given:64x2-16 First notice that both terms have 16 in common. So start by factoring out a 16. 16(4x2 - 1) The inside of the parentheses is a perfect square. This is the same as:(2x)2-(1)2 ⇒ Since 22x2-12 = 4x2-1Which is of the form:a2-b2When you see a polynomial that can be arranged in this way, it can be solved using The Difference of Squares Formula:a2 - b2 = (a-b)(a+b) (Step 1) First identify your a and b terms.(2x)2-(1)2Therefore a = 8x and b = 4(a-b)(a+b)(Step 2) Then plug these values into the difference of squares formula.Hence the factored form of 64x2-16 is:(2x-1)(2x+1) which can also be correctly written as (2x+1)(2x-1).Therefore the final answer is: 16(2x-1)(2x+1) WARNING! A polynomial of the form 64x2+16is NOT factor-ablesince a "sum of squares" formulaa2+b2 DOES NOT EXIST!
Factor:18x2-14x-4
Correct Answer: 2(9x+2)(x-1)Given:18x2-14x-4 First notice that 2 can be factored out from every term in the polynomial above. This gives us: 2(9x2 - 7x - 2)The inside can be factored further:9x2 - 7x - 2 -- This is of the form: Ax2+Bx+C When you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ )(Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses.( __ ? __ )( __ ? __ )How?:Look at the last sign:9x2 - 7x - 2if the last sign is (-), as it is here, then the signs alternate: ( __ - __ )( __ + __ )NOTICE:If it is (+), the two signs will be the same.Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ).(Step 2) Next determine the first and last numbers for each parentheses.( ? - ? )( ? + ? )How?List the factors of the first and last terms:9x2 - 7x - 2 Factors of 9x2 Factors of 2 Multiply Across x 2 2x 9x 1 9x Since the last sign, as we determined above, is (-), the signs are different in either parenthesesand therefore the factors in the above table must SUBTRACT to the middle term of 7x.And since 9x - 2x = 7x, we have picked the correct factors of the first and last terms.The factors of the first and last terms are arranged in the parentheses in this way every time:( # +/- # )( # +/- # )Now look to see if there is a (+) or (-) sign in the middle:9x2 - 7x - 2The middle term is (-), and since the A term is 9, we need to perform a "check"before we know where to place the (-) sign.If we place the (-) sign in the first parentheses:(9x-2)(x+1)Then Check:(1) Multiply the inner terms: -2 * x = -2x(2) Multiply the outer terms: 9x * (+1) = 9xThen add -2x+9x = +7x* Therefore this is wrong; our middle term is NOT (+)If we place the (-) sign in the second parentheses:(9x+ 2)(x-1)Then Check:(1)Multiply the inner terms: +2 * (x) = +2x(2) Multiply the outer terms: 9x * (-1) = -9xThen add +2x-9x = -7x* This IS our middle term, therefore this is correct!Therefore the factored form of 9x2 - 7x - 2 is:(9x+2)(x-1), which can also be written (x-1)(9x+2). And our final answer is: 2(9x+2)(x-1) or 2(x-1)(9x+2)
Factor:3x2-6x-9
Correct Answer: 3(x-3)(x+1)Given:3x2-6x-9 First notice that each of the terms above have 3 in common. So start by factoring 3 out of the polynomial. 3(x2 - 2x - 3) Now we can further factor the inside, it's the same as:x2-2x-3 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ )(Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses.( __ ? __ )( __ ? __ )How?:Look at the last sign:x2-2x-3if the last sign is (-), as it is here, then the signs alternate: ( __ - __ )( __ + __ )NOTICE:If it is (+), the two signs will be the same.Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ).(Step 2) Next determine the first and last numbers for each parentheses.( ? - ? )( ? + ? )How?List the factors of the first and last terms:x2-2x-3 Factors of x2 Factors of 3 Multiply Across x 3 3x x 1 x Since the last sign, as we determined above, is (-), the signs are different in either parenthesesand therefore the factors in the above table must SUBTRACT to the middle term of 6x.And since 3x - x = 2x, we have picked the correct factors of the first and last terms.The factors of the first and last terms are arranged in the parentheses in this way every time:( # +/- # )( # +/- # )Now look to see if there is a (+) or (-) sign in the middle:x2-2x-3The middle term is (-), and since the A term is 3, we need to perform a "check"before we know where to place the (-) sign.If we place the (-) sign in the first parentheses:(x-3)(x+1)Then Check:(1) Multiply the inner terms: -3 * x = -3x(2) Multiply the outer terms: x * (+1) = xThen add -3x+x = -2x* This IS our middle term, therefore this is correct!Therefore our solution is:3(x-3)(x+1), which can also be written 3(x+1)(x-3).
Factor:36x2-4
Correct Answer: 4(3x-1)(3x+1) Given:36x2-4 First notice that both terms have 4 in common. So start by factoring out 4 from both terms. 4(9x2 - 1) Now you can factor the terms within the parentheses:This is the same as:(3x)2-(1)2 ⇒ Since 32x2-12 = 9x2-1Which is of the form:a2-b2When you see a polynomial that can be arranged in this way, it can be solved using The Difference of Squares Formula:a2 - b2 = (a-b)(a+b) (1) First identify your a and b terms.(3x)2-(1)2Therefore a = 3x and b = 1(a-b)(a+b)(2) Then plug these values into the difference of squares formula.Hence the factored form of 36x2-4 is:(3x-1)(3x+1) which can also be correctly written as (3x+1)(3x-1). Therefore the correct solution is: 4(3x-1)(3x+1) WARNING!A polynomial of the form 36x2+4is NOT factor-ablesince a "sum of squares" formulaa2+b2 DOES NOT EXIST!
Factor:12x2-54x+42
Correct Answer: 6(x-1)(2x-7)Given:12x2-54x+42 First notice that 6 is in common with all three terms above. So start by factoring out a 6. 6(2x2 - 9x + 7) Now look at the inside of the parentheses, this can be factored further. This is the same as:2x2-9x+7 -- This is of the form:Ax2+Bx+CWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ )(Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses.( __ ? __ )( __ ? __ )How?:Look at the last sign:2x2-9x+7If it is (+) as in our case, the two signs will be the same.Either ( __ + __ )( __ + __ ) or ( __ - __ )( __ - __ ).Next look at the middle sign:If the middle sign is (-) as in our case, the signs are both (-). ( __ - __ )( __ - __ ).NOTICE:if the last sign was (-) then the signs would alternate regardless of the middle sign,like this: ( __ - __ )( __ + __ )(Step 2) Next determine the first and last numbers for each parentheses.( ? - ? )( ? - ? )How?List the factors of the first and last terms:2x2-9x+7 Factors of 2x2 Factors of 7 Multiply Across x 7 7x 2x 1 2x Since the last sign, as we determined above, is (+), the signs are the same in either parenthesesand therefore the factors in the above table must ADD to the middle term of 9x.And since 7x + 2x = 9x, we have picked the correct factors of the first and last terms.The factors of the first and last terms are arranged in the parentheses in this way every time:( # +/- # )( # +/- # )Therefore our solution is:6(x-1)(2x-7), which can also be written 6(2x-7)(x-1) or in any order because of the commutative property of multiplication.
Factor:7x2-7
Correct Answer: 7(x-1)(x+1)Given:7x2-7First you may notice that both terms have the common value of 7. Start by factoring this out of both terms: 7(x2-1) Now you can factor the x2-1 This is the same as: x2+0x-1 -- This is of the form:Ax2+Bx+C In this case the B term is 0, but you can still factor x2-1 like any other polynomialWhen you see this form,Begin to factor by drawing two parentheses( ___ _ ___ )( ___ _ ___ )(Step 1) Determine the sign (+ or -) that goes in the middle of each parentheses.( __ ? __ )( __ ? __ )How?:Look at the last sign:x2-1if the last sign is (-), as it is here, then the signs alternate: ( __ - __ )( __ + __ )(Step 2) Next determine the first and last numbers for each parentheses.( ? - ? )( ? + ? )How?List the factors of the first and last terms:x2-1 Factors of x2 Factors of 1 Multiply Across x 1 x x 1 x Since the last sign, as we determined above, is (-), the signs are different in either parenthesesand therefore the factors in the above table must SUBTRACT to the middle term of 0.And since x - x = 0, we have picked the correct factors of the first and last terms.The factors of the first and last terms are arranged in the parentheses in this way every time:( # +/- # )( # +/- # ) This gives us (x-1)(x+1) as the factor of x2-1 BUT don't forget the 7 that we factored out in the very beginning!! Final Answer: 7(x-1)(x+1) ------------------------------------- Notice: 7(x-1)(x+1) can be written in any order because the comutative property of multiplication. This means that 7(x-1)(x+1) = 7(x+1)(x-1) = (x-1)7(x+1) = (x+1)7(x-1) = (x-1)(x+1)7 ... etc.
Factor:81x2-9
Correct Answer: 9(3x-1)(3x+1)Given:81x2-9 First notice that both terms have 9 common. So start by factoring out a 9 from both terms. 9(9x2 - 1) The inside of the parentheses can be factored further. 9x2 - 1 is the same as: (3x)2-(1)2 ⇒ Since 32x2-12 = 9x2-1Which is of the form:a2-b2When you see a polynomial that can be arranged in this way, it can be solved using The Difference of Squares Formula:a2 - b2 = (a-b)(a+b) (1) First identify your a and b terms.(3x)2-(1)2Therefore a = 9x and b = 3(a-b)(a+b)(2) Then plug these values into the difference of squares formula.Hence the factored form of 81x2-9 is:(3x-1)(3x+1) which can also be correctly written as (3x+1)(3x-1). Therefore the final answer is: 9(3x-1)(3x+1) WARNING!A polynomial of the form 81x2+9is NOT factor-ablesince a "sum of squares" formulaa2+b2 DOES NOT EXIST!
Simplify:a9b4a5c-6b-3a-2
Correct Answer: Given:a9b4a5c-6b-3a-2 (Step 1): Identify your like terms a9b4a5c-6b-3a-2 (Step 2): Then add their exponents. a9+5+(-2)b4+(-3) c-6When ever two like terms are multiplied together, ADD their powers. Thus we have: a12b1c-6 (Step 3): Move negative powers into the denominator.This changes the power to a positive number - which is required when fully simplifying a mathematical expression. Therefore the final answer is:
Simplify:x2w4x-2y-8w-5x-8
Correct Answer: Given:x2w4x-2y-8w-5x-8 (Step 1): Identify your like terms x2w4x-2y-8w-5x-8 (Step 2): Then add their exponents. x2+(-2)+(-8)w4+(-5) y-8When ever two like terms are multiplied together, ADD their powers. Thus we have: x-8w-1y-8 (Step 3): Move negative powers into the denominator.This changes the power to a positive number - which is required when fully simplifying a mathematical expression. Therefore the final answer is:
Simplify:a7a4a
Correct Answer: a12Given:a7a4a (Step 1): Identify your like termsa7a4a1 All of the terms like terms because they have the same base: a. Notice how I added a power 1 to the last term. When no power is shown, this means there is an understood 1 there. (Step 2): Then add their exponents. a7+4+1 When ever two like terms are multiplied together, ADD their powers.Therefore the final answer is: a12
Simplify: x-1w10x5y9w5x7
Correct Answer: w15x11y9 Given:x-1w10x5y9w5x7 (Step 1): Identify your like termsx-1w10x5y9w5x7 (Step 2): Then add their exponents. w10+5x(-1)+5+7 y9When ever two like terms are multiplied together, ADD their powers. Also, typically the letters are arranged in alphabetical order Thus we have: w15x11y9 Therefore the final answer is: w15x11y9
Simplify:x4y7x6y4
Correct Answer: x10y11Given:x4y7x6y4 (Step 1): Identify your like termsx4y7x6y4 (Step 2): Then add their exponents. x4+6y7+4 When ever two like terms are multiplied together, ADD their powers.Therefore the final answer is: x10y11
Simplify:x2y9x3y3
Correct Answer: x5y12Given:x2y9x3y3 (Step 1): Identify your like termsx2y9x3y3 (Step 2): Then add their exponents. x2+3y9+3 When ever two like terms are multiplied together, ADD their powers.Therefore the final answer is: x5y12
