Objective

 SWBAT review Chapter 9 concepts

Add Polynomials Like Terms

terms that have the same variable

(2x³ – 5x² + x) + (2x² + x³ – 1)

You can add polynomials using the vertical or horizontal format.

Vertical Format

2x³ – 5x² + x x³ + 2x² – 1 3x³ – 3x² + x – 1

Horizontal Format

( 2x³ + x³ ) + ( 2x² – 5x² ) + x – 1 3x³ – 3x² + x – 1

Subtract Polynomials

Like Terms

terms that have the same variable

(4n² + 5) – (-2n² + 2n – 4)

You can subtract polynomials using the vertical or horizontal format.

Vertical Format

4n² + 5

Horizontal Format

( 4n² + 2n² ) – 2n + ( 5 + 4 ) 6n² – 2n + 9 6n² – 2n + 9

Section 9.2 “Multiply Polynomials” When multiplying polynomials use the distributive property. Distribute and multiply each term of the polynomials. Then simply.

2x³ (x³ + 3x² - 2x + 5)

2

x

6  6

x

5  4

x

4  10

x

3 2

x

6  6

x

5  4

x

4  10

x

3

“Multiply Using FOIL” When multiplying a binomial and another First Outer Inner Last

“Multiply Using FOIL” (3a + 4) (a – 2)

3

a

2  6

a

 4

a

 8

combine like terms

3

a

2  2

a

 8

Section 9.4 “Solve Polynomial Equations in Factored Form”

Zero-Product Property

If ab = 0, then a = 0 or b = 0.

The zero-product property is used to solve an equation when one side of the equation is ZERO and the other side is the product of polynomial factors. The solutions of such an equation are called

ROOTS.

(x – 4)(x + 2) x – 4 = 0 x = 4 = 0 x + 2 x = -2 = 0

“Solving Equations By Factoring” 2x² + 8x = 0 When using the zero-product property, sometimes you may need to factor the polynomial, or write it as a product of other polynomials. Look for the greatest common factor (GCF) of the polynomial’s terms. GCF the monomial that divides evenly into EACH term of the polynomial. GCF 2

x

2

x

2  2 

x



x

8

x

 2  2  2 

x

Look for common terms 2

x

2  8

x

 0 2

x

2

x

2

x

(

x

 4 )  0

Solve Equations By Factoring 2x² + 8x = 0 Factor left side of equation 2x x = = 2x(x + 4) = 0 0 0 Zero product property x + 4 = 0 x = -4 The solutions of the equation are 0 and -4.

Section 9.5 “Factor x² + bx + c”

Factoring x² + bx + c

x² + bx + c = (x + p)(x + q)

provided p + q = b and pq = c

x² + 5x + 6 = (x + 3)(x + 2)

Remember FOIL

Factoring polynomials

n² – 6n + 8 Find two ‘negative’ factors of 8 whose sum is -6.

‘-’ factors of 8 -8, -1 -2, -4 Sum of factors -8 + (-1)= -9 -2 + (-4) = -6

(n – 2)(n – 4)

n

2  4

n

 2

n n

2  6

n

 8  8

A shortcut… for factoring ax² + bx + c  (1) multiply ‘a’ and ‘c’ together.

 (2) factor new polynomial normally.

 (3) divide number terms of binomial by ‘a’.

 (4) simplify fractions.

 (5) move denominator of fractions in front of variable terms of binomials

Section 9.6 “Factor ax² + bx + c” Factors of 2 1, 2 1, 2 2x² – 7x + 3 Factors of 3 1, 3 3, 1 Possible factorization (x – 1)(2x – 3) (x – 3)(2x – 1)

First look at the signs of b and c.

Middle term when multiplied -3x – 2x = -5x -x – 6x = -7x

(x – 3)(2x – 1)

2

x

2 

x

 6

x

 3 2

x

2  7

x

 3

Section 9.7 “Factor Special Products”

You can use the following special products patterns to help you factor certain polynomials.

Perfect Square Trinomial Pattern (addition)

a² + 2ab + b² (a + b)² (a + b)(a + b)

Perfect Square Trinomial Pattern (subtraction)

a² – 2ab + b² (a – b)² (a - b)(a - b)

Difference of Two Squares Pattern

a² – b² (a + b) (a – b)

Factoring Polynomials Completely  (1) Factor out greatest common monomial factor.

3x² + 6x = 3x(x + 2)   (2) Look for difference of two squares or perfect square trinomial.

x² + 4x + 4 = (x + 2)(x + 2) 16x² – 49 = (4x + 7)(4x – 7) (3) Factor a trinomial of the form ax ² + bx + c into binomial factors.

3x² – 5x – 2 = (3x + 1)(x – 2)  (4) Factor a polynomial with four terms by grouping.

-4x² + x + x³ - 4 = (x² + 1)(x – 4)