Regents Review #1

Expressions &

Equations(x – 4)(2x + 5)

3x3 – 4x2 + 2x – 1

(4a – 9) – (7a2 + 5a + 9)

4x2 + 8x + 1 = 0 (x – 5) 2 = 25

10x3

5x5 x – 3 = 2x 5x 7

Exponential Expressions

Exponential Expressions

1) xy0 2) (2x2y)(4xy3) 3)

x(1) x

8x3y4

3xy3

Any nonzero number raised to the zero power equals 1.

Multiply coefficients and add exponents.

Divide coefficients and subtract exponents.

2

52

2xyy6x

PolynomialsA polynomial is a sum of terms. Each term is separated by either a + or – sign.

PolynomialsThe degree of a polynomial with one variable is the highest power to which the variable is raised.

PolynomialsA polynomial is written in standard form when the degrees (exponents) are listed from highest to lowest.

PolynomialsWhen adding polynomials, combine like terms.

1) Represent the perimeter of a rectangle as a simplified polynomial expression if the width is 3x – 2 and the length is 2x2 – x + 11.

3x – 2 3x – 2

2x2 – x + 11

2x2 – x + 11

(3x – 2) + (3x – 2) + (2x2 – x + 11) + (2x2 – x + 11)

2x2 + 2x2 + 3x + 3x – x – x – 2 – 2 + 11 + 11

4x2 + 4x + 18

The expression can also be simplified this way.2(3x – 2) + 2(2x2 – x + 11)

PolynomialsWhen subtracting polynomials, distribute the minus sign before combining like terms.

2) Subtract 5x2 – 2y from 12x2 – 5y

(12x2 – 5y) – (5x2 – 2y)

FROM COMES FIRST

12x2 – 5y – 5x2 + 2y

12x2 – 5x2 – 5y + 2y

7x2 – 3y

PolynomialsWhen multiplying polynomials, distribute each term from one set of parentheses to every term in the other set of parentheses.

3) (3x – 4)2

4) Express the area of the rectangle as a simplified polynomial expression.

9x2 – 12x – 12x + 16 9x2 – 24x + 16

2x2 – 4x + 1

2x2 -4x 1

x 2x3 -4x2 x

5 10x2 -20x 5

2x3 + 6x2 – 19x + 5x + 5

(3x – 4)(3x – 4) Expand 3x - 4

PolynomialsWhen dividing polynomials, each term in the numerator is divided by the monomial that appears in the denominator.

42x

2

2

2

3

2

23

3x12x

3x6x

3x12x6x5)

Factoring Polynomials

What does it mean to factor?Create an equivalent expression that is

a “multiplication problem”.

Remember to always factor completely.Factor until you cannot factor anymore!

Factoring

1) Factor out the GCF

2) AM factoring

3) DOTS

1)2x(2x2x4x2

2)3)(x(x65xx2

)4y)(3x4y(3x

16y9x22

42

“Go to Methods”

Binomial difference of two perfect squares

Trinomial ax2 + bx + c , a = 1

FactoringWhat about ax2 + bx + c when a 1?

Factor 5n2 + 9n – 2

5n2 + 10n – 1n – 2

5n(n + 2) – 1(n + 2) (5n – 1)(n + 2)

GCF is 1, Factor by Grouping!

Find two numbers whose product = ac and whose sum = b

ac = (5)(-2) = -10 b = 9 The numbers are -1 and 10

Rewrite the polynomial with 4 terms

Factor out the GCF of each group

Write the factors as 2 binomials

Create two groups5n2 + 10n – 1n – 2

FactoringWhen factoring completely, factor until you cannot factor anymore! 12x10x2x 23 1) 44 81y16x 2)

)9y3y)(4x3y)(2x(2x 22

)9y)(4x9y(4x 2222 6)5x2x(x2

2)3)(x2x(x

The factored form of the polynomial expression is equivalent to the standard form of the polynomial expression.

EquationsWhat types of equations (in one variable) do we need to know how to solve?

1) Equations with rational expressions (fractions)2) Quadratic Equations3) Square Root Equations4) Literal Equations (solving for another variable)

Remember: When solving any type of equation, always use properties of equality and check solution(s).

Rational Equations (proportions)

5(3x – 2) = 10(x + 3)15x – 10 = 10x + 30 5x – 10 = 30 5x = 40 x = 8

Always check solution(s) to any equation

1023x

53x

511

511

1022

511

102)3(

53

88

Rational EquationsHow do we solve a rational equation with more than one fraction?

Option 1: Combine fractions and create a proportion

Option 2: Multiply by the LCD (least common denominator)

7103x

55x

Example: Solve for x.

Rational Equations

7103x

55x

7103x

55x

22

FOO

7103x

10102x

17

10105x

7103x

55x

(7)103x

55x

101010

12x

70105x703x102x

10(7)1(3x)5)2(x

5x + 10 = 70

Create a Proportion Multiply by the LCD

Quadratic Equations1) ax2 + c = 0 Ex: 2x2 – 32 = 0 2x2 = 32 x2 = 16 x = x = 4 or x = {4,-4}

16

2) ax2 + bx + c = 0Ex: x2 – 5x = -6

Isolate x2 and take the square root.

Set the equation equal to zero.Factor.Set each factor equal to zero and solve.Zero Product Property

x2 – 5x + 6 = 0

(x – 2)(x – 3)= 0

x – 2 = 0 x – 3 = 0 x = 2 x = 3 x = {2,3}

Quadratic EquationsHow do we solve a quadratic equation that cannot be factored?

Example: Find the roots of x2 – 2x – 5 = 0.

)1(2

)5)(1(4)2()2( 2 x

2

242x

2

622x

1

611x

}61,6{1 x

61x

Use the quadratic formula:a

acbbx

2

42

a = 1, b = -2, c = -5

626424

Quadratic EquationsThe equation can also be solved by completing the square.Find the roots of x2 – 2x – 5 = 0.

x2 – 2x – 5 = 0x2 – 2x = 5x2 – 2x _____ = 5 _______x2 – 2x + 1 = 5 + 1

(x – 1)(x – 1)

1)1(2

2

22

22

b

(x – 1)2 = 6

61)(x 2

61x61x

Square Root Equations Solve 3193x5

403x5

83x

2283x 643x

321.x

Isolate the

Square both sides of the equation to eliminate the

Literal EquationsWhen solving literal equations, isolate the indicated variable using properties of equality.

candxa,oftermsinyforSolve

cyxay

xac

xax)y(a

xac

y

y(a + x) = cFactor out the variable that you are solving for.

Literal Equations

bh21

A

bh2A

bbh

b2A

hb

2A

bandAoftermsinhforSolve

Multiply both sides of the equation by 2 to eliminate the fractional coefficient.

Now it’s your turn to review on your own!

Using the information presented today and the study guide posted on halgebra.org,

complete the practice problem set.

Regents Review #2 Friday, May 15th

Be there!