Excelsior Algebra I Unit 6
Factor the expression: 8x^2 - 2x - 1
The answer is (2x - 1) (4x + 1) To solve, you need to factor. Check to see if there is anything in common between all of the terms first. In this case, there isn't any common factors. Next, you need to find the factors of 8x^2 -2x - 1. To do this, you will need to find the combination of factors of 8 and -1 that SUBTRACT to -2. These are (2x - 1) (4x + 1). I use those in my answer to get (2x - 1) (4x + 1).
Factor the expression: 9x^2 - 64y^2
The answer is (3x + 8y)(3x - 8y) To solve, you need to factor. Check to see if there is anything in common between all of the terms first. In this case, there isn't any common factors. Next, you need to recognize that since there is only two terms, and they have nothing in common, in order to factor, it needs to be a difference a two squares. These have perfect squares for each term and are being subtracted. To factor, take the square root of each term and then use those as the terms inside each binomial with one having a plus sign and the other a minus sign. (3x + 8y)(3x - 8y)
Factor the expression: 4x^3 + 4x^2 + 3x + 3
The answer is (4x^2 + 3) (x + 1) To solve, you need to factor. Check to see if there is anything in common between all of the terms first. In this case, there isn't any common factors. Next, I hope that you notice that there are four terms to this problem. When this happens, if there are no common terms in the whole problem, separate it into two parts. The first part is 4x^3 + 4x^2 and the second part is 3x + 3. Find any common factors for each section as follows 4x^3 + 4x^2: 4x^2(x + 1) 3x + 3: 3(x + 1) Notice that each section has an (x + 1) as part of their factors. Combine the parts of each section other than (x + 1) to get (4x^2 + 3) along with the (x + 1) to get the answer: (4x^2 + 3)(x + 1)
Factor the expression: 36x^2 + 48x +16
The answer is (6x + 4)^2 To solve, you need to factor. Check to see if there is anything in common between all of the terms first. In this case, there isn't any common factors. Next, you need to find the factors of 36x^2 + 48x +16. To do this, you will need to find the combination of factors of 36 and 16 that ADD to +48. These are (6x + 4) (6x + 4). I use those to simplify my answer to get (6x + 4)^2.
Factor the expression: 21x^2 - 4x - 1
The answer is (7x + 1) (3x - 1) To solve, you need to factor. Check to see if there is anything in common between all of the terms first. In this case, there isn't any common factors. Next, you need to find the factors of 21x^2 - 4x - 1. To do this, you will need to find the combination of factors of 21 and -1 that SUBTRACT to -4. These are (7x + 1) (3x - 1). I use those in my answer to get (7x + 1) (3x - 1).
Factor the expression: x^2 + 17x + 66
The answer is (x + 6) (x + 11) To solve, you need to factor. Check to see if there is anything in common between all of the terms first. In this case, there isn't so we will now need to find factors of 66 that ADD up to +17. These factors are +6 and +11. I use those in my answer to get (x + 6) (x + 11).
Factor: x^2 - x - 56
The answer is (x - 8) (x + 7) To solve, you need to factor. Check to see if there is anything in common between all of the terms first. In this case, there isn't so we will now need to find factors of 56 that SUBTRACT to -1. These factors are -8 and +7. I use those in my answer to get (x - 8) (x + 7).
Factor: x^2 - 9x + 8
The answer is (x - 8) (x - 1) To solve, you need to factor. Check to see if there is anything in common between all of the terms first. In this case, there isn't so we will now need to find factors of 8 that ADD up to -9. These factors are -8 and -1. I use those in my answer to get (x - 8) (x - 1).
Simplify the sum: (7g^3 + 7g^2 + 5) + (3g^3 - 4g + 11)
The answer is 10g^3 + 7g^2 - 4g + 16 When adding polynomials, you you can only combine like terms (terms with the EXACT SAME variable and exponent) and you keep the variable. Your answers should be in standard form. 7g^3 + 3g^3 = 10g^3 7g^2 has no like term -4g has no like term 5 + 11 = 16
Simplify the product: (3p + 5) (4p + 2)
The answer is 12p^2 + 26p + 10 To solve this, you need to FOIL. FOIL stands for FIRST, OUTSIDE, INSIDE, LAST. Multiply the FIRST term of each binomial: 3p (4p) = 12p^2 Multiply the OUTSIDE terms: 3p (2) = 6p Multiply the INSIDE terms: 5 (4p) = 20p Multiply the LAST term of each binomial: 5 (2) = 10 Normally, the two middle terms will be like and can be combined 12p^2 + 6p + 20p + 10 = 12p^2 + 26p +10
Simplify the product: 4w^5 (3w^5 + 8w^3 + 7w + 2)
The answer is 12w^10 + 32w^8 + 28w^6 + 8w^5 When multiplying polynomials, you will distribute the 3h to all of the terms inside the parenthesis. Remember that when you multiply variables, you add the exponents. 4w^5 (3w^5) = 12w^10 4w^5 (8w^3) = 32w^8 4w^5 (7w) = 28w^6 4w^5 (2) = 8w^5
Factor the expression completely: 16x^2 - 28x - 30
The answer is 2(4x - 3)(2x + 5) To solve, you need to factor. Check to see if there is anything in common between all of the terms first. In this case, there is a common factor of a 2. So, we need to factor out a 2 first. Next, you need to find the factors of 8x^2 -14x - 15. To do this, you will need to find the combination of factors of 8 and 15 that SUBTRACT to -14. These are (4x - 3)(2x + 5). When combined with our common factor, the answer is 2(4x - 3)(2x + 5).
Find the product: (5x^2 - 4)(5x^2 + 4)
The answer is 25x^4 - 16 To solve this, you will FOIL which stands for FIRST, OUTSIDE, INSIDE, LAST. Multiply the FIRST term of each binomial: 5x^2 (5x^2) = 25x^4 Multiply the OUTSIDE terms: 5x^2 (4) = 20x^2 Multiply the INSIDE terms: (-4) (5x^2) = -20x^2 Multiply the LAST term of each binomial: (-4) (4) = -16 Normally, the two middle terms will be like and can be combined 25x^4 + 20x^2 - 20x^2 - 16 = 25x^4 - 16
Factor the polynomial: 3b^3 + 6b^2 + 12b
The answer is 3b (b^2 + 2b + 4) When factoring, always look for a common term that you can factor out of each term first. In this case, each term has a 3 and a b that can taken out of each term. You always write down the common factor and then what is left afterwards. 3b^3 / 3b = b^2 6b^2 / 3b = 2b 12b / 3b = 4
Simplify the product: 3h (h^2 + 4h + 6)
The answer is 3h^3 + 12h^2 + 18h When multiplying polynomials, you will distribute the 3h to all of the terms inside the parenthesis. Remember that when you multiply variables, you add the exponents. 3h (h^2) = 3h^3 3h (4h) = 12h^2 3h (6) = 18h
Find the square: (7x + 8)^2
The answer is 49x^2 + 112x + 64 To solve this, remember that (7x + 8)^2 means (7x + 8)(7x + 8). Next, you will FOIL which stands for FIRST, OUTSIDE, INSIDE, LAST. Multiply the FIRST term of each binomial: 7x (7x) = 49x^2 Multiply the OUTSIDE terms: 7x (8) = 56x Multiply the INSIDE terms: 8 (7x) = 56x Multiply the LAST term of each binomial: 8 (8) = 64 Normally, the two middle terms will be like and can be combined 49x^2 + 56x + 56x + 64 = 49x^2 + 112x + 64
Simplify: (2r + 4)(2r^2 - 5r - 2)
The answer is 4r^3 - 2r^2 - 24r - 8 To solve this, multiply 2r by EVERYTHING in the second parenthesis, then multiply 4 by EVERYTHING in the second parenthesis. Combine any like terms. 2r (2r^2 - 5r - 2) = 4r^3 - 10r^2 - 4r 4 (2r^2 - 5r - 2) = 8r^2 - 20r - 8 4r^3 - 10r^2 - 4r + 8r^2 - 20r - 8 = 4r^3 - 2r^2 - 24r - 8
Factor the expression: 4x^4 - 12x^3 - 24x^2 + 72x
The answer is 4x (x^2 - 6) (x - 3) To solve, you need to factor. Check to see if there is anything in common between all of the terms first. In this case, there is a 4x in common any common factors. When you factor out a 4x from everything, what you get is: 4x(x^3 - 3x^2 - 6x +18) Next, I hope that you notice that there are four terms left inside the parenthesis of this problem. When this happens, separate it into two parts. The first part is x^3 - 3x^2 and the second part is - 6x +18. Find any common factors for each section as follows x^3 - 3x^2: x^2 (x - 3) - 6x +18: -6 (x - 3) Notice that each section has an (x - 3) as part of their factors. Combine the parts of each section other than (x - 3) to get (x^2 - 6) along with the (x - 3) and the original GCD 4x to get the answer: 4x (x^2 - 6) (x - 3) 2x (2x^3(x - 3) - 12x (x - 3)
Simplify the difference: (-6v - 4v^4 + 2) - (-9v^4 - 4 - 3v)
The answer is 5v^4 - 3v + 6 When subtracting polynomials, you can only combine like terms (terms with the EXACT SAME variable and exponent) and you keep the variable. Your answers should be in standard form. -4v^4 - (-9v^4) = 5v^4 because negative negatives are positive and 9 - 4 = 5 -6v - (-3v) = -3v because negative negatives are positive and -6 + 3 = -3 2 - (-4) = 6 because negative negatives are positive and 2 + 4 = 6
Find the GCF of the terms of the polynomial 7j^8 + 28j^4
The answer is 7j^4 To find the GCF, break down each term into its simplest terms possible: 7j^4: 7 • j • j • j • j • j • j • j • j 28j^4: 2 • 2 • 7 • j • j • j • j The terms they have in common is 7j^4
Complete: x^2 + 14x + 48 = (x + 6) (x + ____)
The answer is 8 To solve, you need to find the factors of 48 that ADD up to +14. These are +6 and + 8. Since they already gave you +6, the answer is +8
Find the square: (3x - 5)^2
The answer is 9x^2 - 30x + 25 To solve this, remember that (3x - 5)^2 means (3x - 5)(3x - 5). Next, you will FOIL which stands for FIRST, OUTSIDE, INSIDE, LAST. Multiply the FIRST term of each binomial: 3x (3x) = 9x^2 Multiply the OUTSIDE terms: 3x (-5) = -15x Multiply the INSIDE terms: (-5) (3x) = -15x Multiply the LAST term of each binomial: (-5)(-5) = 25 Normally, the two middle terms will be like and can be combined 9x^2 -15x - 15x + 25 = 6x^2 - 30x + 25
Find the product: (3x + 2) (3x - 2)
The answer is 9x^2 - 4 To solve this, you will FOIL which stands for FIRST, OUTSIDE, INSIDE, LAST. Multiply the FIRST term of each binomial: 3x (3x) = 9x^2 Multiply the OUTSIDE terms: 3x ( -2) = -6x Multiply the INSIDE terms: 2 (3x) = 6x Multiply the LAST term of each binomial: 2 (-2) = -4 Normally, the two middle terms will be like and can be combined 9x^2 - 6x + 6x - 4 = 9x^2 - 4
Find the degree of the monomial 16m^4 n^8
To find the degree of a monomial, add all of the exponents of the variables together. The answer is 12
Write the polynomial in standard form 5q - q^3 + 17q^2 - 2
To put this in standard form, the exponents should go from the highest number to the lowest number and any terms without a variable at all (just a number) will go at the very end. - q^3 + 17g^2 + 5q - 2
