Algebra 1

Special Binomial Multiplication

(4x-3)² (5x+2)(5x-2) (4x-3)(4x-3) 25x²-10x+10x-4 16x²-12x-12x+9 25x²-4 16x²-24x+9

Associative Property of Addition

(a+b)+c = a+(b+c)

Associative Property of Multiplication

(ab)c = a(bc)

Multiply in Binomial

(x+5)(x-1) x²-1x+5x-5 x²+4x-5

7a-4(b+2) if; a=5 b=2

19

3x+2x-x+2x^2

2x^2+4x

(6+25-7)÷6

4

Inverse Variation

A negative or positive constant rate linear function that avoids the origins. Equation is y=k/x (k = yx) is the constant of the proportionality. Ex. y=3/x or y=-2/x.

Function

A relation is a function if and only if the element in the domain is paired with a unique output.

f(x)=1/2x+3

f(x)=1/2x+3; plug in any number in for x and solve. What you plug in for x is your Domain, and what you get for your y is your Range.

One solutions

two lines that intersect at one point

Which one is a Function? { (5,6)(3,2)(-3,2)(4,1) } { (5,6)(1,4)(3,4)(1,-2) }

{ (5,6)(3,2)(-3,2)(4,1) }

√392

√196*√2=14√2

Distributing radicals

√5(√10+4√3= √50+4√15=√25√2+4√25= 5√2+4√15

4y^2-2y+6y^3-16y-10y^2+8y^3

+8y^3+4y^2-16y+6y^3-10y^2-2y= 14y^3-6y^2-18y

-4(8x-4)+3x

-29x+16

-(1-5n)-7n

-2n-1

Elimination Method

-8x-10y=28 1. Line up term 4x+10y=-24 2. Opposite coefficients -4x=4 3. Add each term -4x/-4=4/-4 4. Substitute into original equation x=-1 -8x-10y=28 (-1.-2) -8(-1)-10y=28 8-10y=28 -8 -8 -10y=20 -10y/-10=20/-10 y=-2

x⁰

1

Cubed Roots (Perfect Cubes)

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859,8000, 9261, 10648, 12167, 13824, 15,625. (This 1-25).

2/3x=24

1. (3/2) 2/3x=24(3/2) 2. x= 3(12) 3. x=36

(-20)(x+3/4)

1. (4)(-20)(x+3/4)(4) 2. -80=x+3 3. -83=x

Distributive Property (Algebraically)

1. 3(x+8) 2. 3x+3(8) 3. 3x+24

3x+5=38

1. 3x+5=38 -5 -5 2. 3x=33 3. 3x/3=33/3 4. x=11

Distributive Property (Numerically)

1. 7(53) 2. 7(50+3) 3. 7(50)+ 7(3) 4. 350+21 5. 371

a-3=10 (Addition Property)

1. a-3=10 +3 +3 2. a=13

x⁻¹

1/x¹

20÷(4-(10-8))

10

72÷9+7

15

Inequalities

2x+3(3-x) < 14+9 2x+9-3x < 23 -1x+9 < 23 -9 -9 -1x<14 -1x/-1 < 14/-1 x > -14

No Solution

2x+5(x-4)=7x-13 2x+5x-20=7x-13 7x-20=7x-13 +20 +20 7x=7x+7 -7x -7x 0=7

Factoring a polynomial #2

2x²+11x+12 Multiply the outside #´s Factors of 24 24x Bring down outside numbers 8,3 2x²+8x+3x+12 Find GCF of first two seperatly from 2x(x+4)+3(x+4) the last two terms Take the #'s not in ( ) and (2x+3)(x+4) put them in ( )

6x⁸/3x⁵

2x³

Infinite Soultions

3(2x-5)=x+2(2x-7)-1+x 6x-15=x+4x-14-1+x 6x-15=6x-15 +15 +15 6x=6x 6x/6=6x/6 x=x

Find the slope given two points; (4,-2)(-1,3)

3-(-2)/-1-4 5/-5=-1 slope: -1

Evaluating Expressions

3a+7=12 The whole thing is the equation, 3a+7 is the expression, the three is the Coefficient, the a is the variable, and 7 is the Constant.

Find the Slope given the equation; 3x-6y+8=0

3x-6y+8=0 -3x -3x -6y+8=-3x -8 -8 -6y=-3x-8 slope:1/2 -6y/-6=-3x-8/-6 y=1/2x+4/3

2x^2+2x+x^2+x

3x^3+3x

3√12

3√12=3√4*√3=3*2√3=6√3

8+7(7n-4)

49n-20

Determine if the point is a solution of the equation. 5x+2y=14 and (0,7)

5(0)+2(7)=14 0+14=14 14=14 true

Multi step Equation (Distributive Property)

5(4x-1)+2=24; first you need to distribute the 5 out to the 4x and the -1. Your equation will now look like this 20x-5+2=24. Next, you need to add and get rid of the constant. Your equation will now look like this 20x= 27. Then divide the coefficient and the 27 and then you'll have an answer of x=1.35.

One Solution

5x+2(3x-4)=8x+8 5x+6x-8=8x+8 11x-8=8x+8 11x=8x+16 3x=16 x=5 1/3

3x-2y+x^2 if; x=-3 and y=-4

8

Direct Variation

A negative or positive constant rate linear function that has a y-Intercept at (0,0). Equation is y=kx (k = y/x) is the constant of the proportionality. Ex. y=3x or y=-2.

Relations

A relationship between a set of values (ie. x-value and y-value).

Literal Equations

A/L = L*W/L;W (A)=(1/2bh);b A/L=W 2(A)= (2)(1/2bh) 2A=bh 2A/h=b

Operations with Radicals

Add or Subtract: 5√2+3√2-4√3+6√3= 8√2+2√3 √8+3√2+3√27+2√18=2√2+3√2+9√3+6√2= 11√2+9√3 Multiply or Divide: √5*√2= √10 1. √10*√15= 5*2 5*3 = 5√6 2. √14/√8 7*2 4*2 7 4=2√2= √7/2

Addition Symbol

Addition, add, sum, plus, more than, increased by. An examples a number more than 8 = 8+x.

Two Step Equation

An equation having two different operations.

Linear Inequality

An inequality in two variables whose graph is a region of the coordinate plane that is bounded by a line.

Cube Root

Cube Root examples 3√1=1 and 3√64=4

Is this Direct or Inverse Variation?; y=5x

Direct Variation

Division Symbol

Division, Divide, Divisible, Fraction, Quotient, Over, Each, Out of, Ratio. An example a quotient of a number and 7 = x/7

Factorial Symbol

Factorial [consecutive factors (products)]. An examples 6!=6×5×4×3×2×1= 720

A farm house shelters 11 animals. Some are goats and some are ducks. Altogether there are 34 legs. How many of each animal are there?

Figure out x and y. X= # of Goats legs =4 Y= # of Ducks legs = 2 Figure out first equation (The equation that takes the x and y and combines them into a #) x+y=11 Figure out the 2nd equation 4x+2y=34 Use the Substitution or the Elimination method to solve. Your answer should be (6 goats, 5 ducks)

Factoring a polynomial

Find factors that when multiplied equal the constant and when added equal the ¨bx¨. ax²+bx+c. The two inside numbers the ¨bx¨ should equal 3x. x²+3x+2 (x+1)(x+2) x²+1x+2x+2

GCF (Greatest Common Factor)

Finding the GCF of Polynomials. The biggest number that can be multiplied by a number to make the coefficient. Ex. 20a²+ 14a= 2a(10a+7) the 2a is the GCF.

Order of Operations GE(MD)(AS)

G: Grouping, E: Exponents, M: Multiply, D: Divide, A: Add, and S: Subtract. When on the stages of Multiplying, Dividing, Adding, and Subtracting always go from left to right.

Transitive Property of Equality

If a = b and b = c, then a = c

Substitution Property

If a = b+c and b = d then a = d+c

Addition Property of Equality

If a = b, then a+c = b+c

Subtraction Property of Equality

If a = b, then a-c = b-c

Multiplication Property of Equality

If a = b, then ac = bc

Horizontal Slope

If the slope is horizontal then the slope has a 0 as its rise and run. ex. 0/-3 = 0. What this mean is the two points have the same y-value so you would just start at the y-value and make a horizontal line. The slope is always 0.

Vertical Slope

If the slope is vertical then the slope has a 0 as its run. ex. -3/0 = ¨Error¨.What this mean is the two points have the same x-value so you would just start at the x-value and make a vertical line. The slope is always undefined.

Is this Direct or Inverse Variation?;y=5/x

Inverse Variation

Polynomials

Many Termed equation. When adding or subtract terms the exponents stays the same. Ex. 5y²+3y+2y²+6y=7y²+9y

Multiplication Symbol

Multiplication, Multiply, times, product of. An example the product of 6 and a number = 6x or 6(x)

Square Roots (Perfect Squares)

Numbers whose square roots are integers. They are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 566, 625. (This 1-25).

Monomial

One term ex. 2x

no solution

Parallel lines

Law of exponents

Product Rule: Coefficients are not exponents, Multiply coefficients ¨normally¨,Treat each base separately. Ex. a³ * a¹ = a⁴ Quotient Rule: Coefficients are not exponents, Simplify,reduce,divide coefficients(They are allowed to be negative). *Top heavy vs.Bottom Heavy* Ex. -15a⁶/-25a¹¹ = 3/5a⁵ Power Rule: Distribute the outside exponent to each base by multiplying. Even to the coefficients exponent. Ex. (4a⁶)² = 4²a¹²=16a¹² Combo: Ex. 3a²(2a¹)/4a⁸= 6a³/4a⁸=3/2a⁵

Combine Like Terms

Simplify expressions by adding or subtracting like terms. Key things to remember are like terms must have the same variable and they have to be in descending order.

2x+3y=6

Solving for y its 2x+3y=6 -2x -2x 3y=-2x+6 3y/3=-2x+6/ y=-2/3x+2

Square Root Symbol

Square root (x^2), Cubed root (x^3), radical, root. Some examples are √49=7, √121=11, √625=25, and 3√27=9.

Subtraction Symbol

Subtraction, subtract, minus less that decreased by. An examples 5 less than a number = x-5.

Slope using the equation

Take two points of the line. ex. (5,3) (6,1) Then put x_1 and y_1 on the first point and x_2 and y_2. Next, plug the numbers into the equation and the you'll have you're slope.

Absolute Value

The absolute value of a number is its distance from 0 on a number line. An example |-3|=3

Slope (rise/run)

The difference of the y-values divided by the difference of the x-values or the ratio vertical change and horizontal change.

Y=mx+b (Slope intercept form)

The m represents the slope and the b represents the y-intercept..

One Solution Parabola

The vertex is on one point and there are no other intersections.

Inverse Variation Table

The x-values divided by the constant is Inverse Variation. The y-value multiplied by the x-value is Inverse Variation.

Direct Variation Table

The x-values divided by the y-value is Direct Variation. The constant multiplied by the x-value is Direct Variation.

Domain

The x-values of a set - independent or input values. All the x-values have to be unique.

Range

The y-values of a set - dependent or output values.

Two solutions Parabola

There are to point of intersection.

No Solutions Parabola

This has no real solutions.

Trinomial

Three terms. 2x²+7x²+6y=9x²+6y

Infinite Solution

Two line over top of each other

Systems of Equations

Two or more equations that intersect at a single point. Ex. y=2x+5,y=1/2+8. Algebraically y=2x+5 -> 9=2(2)+5= 9=9 y=1/2x+8 -> 9=1/2(2)+8= 9=9.

Binomial

Two terms. ex. 2x+3x=5x

Function on a Graph

Use the line to see if it a function. Draw a vertical line down the graph and if it cross the line once it is a function. If it crosses more than once it is not a function.

Commutative Property of Addition

a + b + c = c + b + a

Reflexive Property

a = a

Line of Best Fit

a line drawn in a scatter plot to fit most of the dots and shows the relationship between the two sets of data

Distributive Property

a(b + c) = ab + ac

Multiplicative Property of Zero

a*0 = 0

Additive Indentity Property

a*1 = a

Additive Inverse Property

a+(-a)= 0

Multiplicative Inverse

a/b*b/a = 1 or 7*1/7=1

Communtative Property of Multiplication

abc = cba

Linear Equation

an equation between two variables that gives a straight line when plotted on a graph.

One Step Equation

an equation that requires only one step to solve...one operation

Quadratic Equations (Find the root)

ax²+bx+c=0 x²+11x+24=0 * Always equal 0* (x+8)(x+3)=0 Factor the Polynomial x+8=0 x+3=0 Set each factor for 0 solve for x -8 -3 x=-8 x=-3 These are the "Zeros""Roots""Solutions""x-Intercept"

Quadratic Formula

ax²+bx+c=0 and x = -b ± √(b² - 4ac)/2a. Ex. x²+7x+12=0 x= (-7)+-√7²-4(1)(12)/2(1) x=-7+-√49-48/2 x= -7+-√1/2 x=-7+-1/2 x=-7+1/2=-3 -3 and -4 are the roots x=-7-1/2=-4

Find the x-intercept and the y-intercept; -20x+40y=-80

for x-Inter.-20x+40y=-80 for y-Inter. -20(0)+40y=-80 40y=-80 -20x+40(0)=-80 40y/40=-80/40 -20x+0=-80 -20x=-80 y=-2 -20x/-20=-80/-20 x=4

Division Property of Equality

if a = b, then a/c = b/c

Symmetric Property of Equality

if a = b, then b = a

Simplifying Expression

to use distribution to combine like terms

Mapping Function x (5 3 -2) x (2 5 -1) x (0 -20 (Which Is A B C a Function?) y (4 5 0) y (3) y (4 6 7)

x (5 3 -2) x (2 5 -1) A B y (4 5 0) y (3)

Table (Which is a Function?) x y x y x y 4 5 2 2 5 -1 3 4 1 2 5 0 2 3 -2 2 5 1 1 4 0 2 5 2

x y x y 4 5 2 2 3 4 1 2 2 3 -2 2 1 4 0 2

Linear Equations Tables

x y y= -x+2 -2 4 -1 3 0 2 0-3/2-(-1)= -3/3 slope: -1 1 1 2 0

√x⁷

x³√x Always divide exponents by 2 2/7= 2*3=6 7-6=1 the number that had to be multiplied to get another number that isn't 2 is the exponent that goes outside. When you subtract the two numbers what left of on the inside of the root. If the is a 0 don't put anything.

(x⁵y⁷)²

x¹⁰y¹⁴

(x²)(x³)(x⁴)

x⁹

Point Slope form

y-y_1=m(x-x_1)

Substitution Method

y=(-2x+1) 2x-2(-2x+1)=4 y=-2(1)+1 2x-2y=4 2x+4x-2=4 y = -2+1 6x-2=4 y=-1 +2 +2 (1,-1) 6x=6 6x/6=6/6 x=1

Function Notation

y=2x+5 -> f(x)=2x=5 x y 3 11 4 3 5 15