Example 1

Factor out a common binomial

Factor the expression.

a. 2x ( x + 4 ) – 3( x + 4 )
b. 3y2 ( y – 2 ) + 5( 2 – y ...
Example 1

Factor out a common binomial

3y2 ( y – 2 ) + 5( 2 – y ) = 3y2 ( y – 2 ) – 5( y – 2) Factor – 1 from
(2 – y).

...
Factor by Grouping
 In a polynomial with 4 terms, factor a common

monomial from pairs of terms, then look for a
common b...
Example 2

Factor by grouping

Factor x3 + 3x2 + x + 3.
x3 + 3x2 + x + 3 = ( x3 + 3x2 ) + ( x + 3)

Group terms.

= x2 ( x...
Example 3

Factor by grouping

Factor x3 – 6 + 2x – 3x2
SOLUTION
The terms x3 and – 6 have no common factor. Use the
commu...
Example 3

Factor by grouping

CHECK
Check your factorization using a graphing calculator.
Graph y1 = x3 – 6 + 2x – 3x2 an...
Factoring Completely
 A factorable polynomial with integer coefficients is

factored completely if it is written as a pro...
Guidelines for Factoring Polynomials Completely
1. Factor out the greatest common monomial factor. (9.5)

3x 2 + 6x = 3x(x...
Example 4

Multiple Choice Practice

Which is the completely factored form of 12n2 + 10n – 8?
2(3n – 4 ) (2n + 1 )

2(3n +...
Example 5
Solve

Solve a polynomial equation

3x3 + 18x2

= – 24x.

SOLUTION
3x3 + 18x2
3x3 + 18x2
3x ( x2 +

+ 24x

= – 2...
Example 5

Solve a polynomial equation

ANSWER
The solutions of the equation are 0, – 2, and – 4.
CHECK
Check each solutio...
Example 6

Solve a multi-step problem

TERRARIUM
A large terrarium is used to display a box turtle in a
pet store. The ter...
Example 6

Solve a multi-step problem

SOLUTION
STEP 1 Write a verbal model. Then write an equation.

8748

= ( w + 36 )

...
Example 6

Solve a multi-step problem

0 = (w3 + 27w2 ) – (324w + 8748 )

Group terms.

0 = w2( w + 27 ) – 324( w + 27 )

...
Example 6

Solve a multi-step problem

STEP 3 Find the length and height.
Length w + 36 = 18 + 36 = 54
Height = w – 9 = 18...
9.9 Warm-Up (Day 1)
Factor the expression.
1.

x(x - 2)+ (x - 2)

2.

a + 3a + a + 3

3.

y + 2x + yx + 2y

3

2

2
9.9 Warm-Up (Day 2)
Factor the expression.
1.

4y -16y

2.

6g - 24g + 24g

3.

h + 4h - 25h -100

6

3

3

4

2

2
9.9
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  • Day 2
  • 1. (x-2)(x+1) 2. (a^2+1)(a+3) 3. (y+x)(y+2)
  • 1. 4y^4(y+2)(y-2) 2. 6g(g-2)^2 3. (h+4)(h-5)(h+5)
  • 9.9

    1. 1. Example 1 Factor out a common binomial Factor the expression. a. 2x ( x + 4 ) – 3( x + 4 ) b. 3y2 ( y – 2 ) + 5( 2 – y ) SOLUTION a. 2x ( x + 4 ) – 3( x + 4 ) = ( 2x – 3 ) ( x + 4 ) Distributive property b. The binomials y – 2 and 2 – y are opposites. Factor –1 from 2 – y to obtain y – 2 as a common binomial factor.
    2. 2. Example 1 Factor out a common binomial 3y2 ( y – 2 ) + 5( 2 – y ) = 3y2 ( y – 2 ) – 5( y – 2) Factor – 1 from (2 – y). = ( 3y2 – 5 ) ( y – 2) Distributive property
    3. 3. Factor by Grouping  In a polynomial with 4 terms, factor a common monomial from pairs of terms, then look for a common binomial factor.
    4. 4. Example 2 Factor by grouping Factor x3 + 3x2 + x + 3. x3 + 3x2 + x + 3 = ( x3 + 3x2 ) + ( x + 3) Group terms. = x2 ( x + 3 ) + 1( x + 3) Factor each group; write x + 3 as 1(x + 3). = ( x2 + 1 ) ( x + 3) Distributive property
    5. 5. Example 3 Factor by grouping Factor x3 – 6 + 2x – 3x2 SOLUTION The terms x3 and – 6 have no common factor. Use the commutative property to rearrange the terms so that you can group terms with a common factor. x3 – 6 + 2x – 3x2 = x3 – 3x2 + 2x – 6 Rearrange terms. = ( x3 – 3x2 ) + ( 2x – 6) Group terms. = x2 ( x – 3 ) + 2( x – 3) Factor each group. = ( x2 + 2 ) ( x – 3) Distributive property
    6. 6. Example 3 Factor by grouping CHECK Check your factorization using a graphing calculator. Graph y1 = x3 – 6 + 2x – 3x2 and y2 = ( x – 3) ( x2 + 2 ). Because the graphs coincide, you know that your factorization is correct.
    7. 7. Factoring Completely  A factorable polynomial with integer coefficients is factored completely if it is written as a product of unfactorable polynomials with integer coefficients.
    8. 8. Guidelines for Factoring Polynomials Completely 1. Factor out the greatest common monomial factor. (9.5) 3x 2 + 6x = 3x(x + 2) 2. Look for a difference of two squares or a perfect square trinomial. (9.8) x 2 + 4x + 4 = (x + 2)2 x - 9 = (x + 3)(x - 3) 3. Factor a trinomial of the form ax 2 + bx + c into a 2 product of binomial factors. (9.6 & 9.7) 3x - 5x - 2 = (3x +1)(x - 2) 2 4. Factor a polynomial with four terms by grouping. (9.9) x3 + x - 4x 2 - 4 = (x - 4)(x 2 +1)
    9. 9. Example 4 Multiple Choice Practice Which is the completely factored form of 12n2 + 10n – 8? 2(3n – 4 ) (2n + 1 ) 2(3n + 4 ) (2n – 1 ) 3(3n – 2 ) (2n + 2 ) 3(6n – 2 ) ( n + 4 ) SOLUTION 12n2 + 10n – 8 = 2(6n2 + 5n – 4 ) = 2(3n + 4 ) (2n – 1 ) ANSWER The correct answer is B. Factor out 2. Factor trinomial.
    10. 10. Example 5 Solve Solve a polynomial equation 3x3 + 18x2 = – 24x. SOLUTION 3x3 + 18x2 3x3 + 18x2 3x ( x2 + + 24x = – 24x Write original equation. = 0 Add 24x to each side. 6x + 8 ) = 0 Factor out 3x. 3x ( x + 2 ) ( x + 4 ) = 0 3x = 0 or x = 0 or x + 2 = 0 or x = – 2 or Factor trinomial. x + 4 = 0 x = –4 Zero-product property Solve for x.
    11. 11. Example 5 Solve a polynomial equation ANSWER The solutions of the equation are 0, – 2, and – 4. CHECK Check each solution by substituting it for x in the equation. One check is shown here. ? ( – 2 )3 + 18( – 2 )2 = – 24 ( – 2 ) 3 ? – 24 + 72 = 48 48 = 48
    12. 12. Example 6 Solve a multi-step problem TERRARIUM A large terrarium is used to display a box turtle in a pet store. The terrarium has the shape of a rectangular prism with a volume of 8748 cubic inches. The dimensions of the terrarium are shown. Find the length, width, and height of the terrarium.
    13. 13. Example 6 Solve a multi-step problem SOLUTION STEP 1 Write a verbal model. Then write an equation. 8748 = ( w + 36 ) • w • (w – 9) STEP 2 Solve the equation for w. 8748 = (w + 36 ) ( w) ( w – 9 ) Write equation. 0 = w3 + 27w2 – 324w – 8748 Multiply. Subtract 8748 from each side.
    14. 14. Example 6 Solve a multi-step problem 0 = (w3 + 27w2 ) – (324w + 8748 ) Group terms. 0 = w2( w + 27 ) – 324( w + 27 ) Factor each group. 0 = (w2 – 324 ) ( w + 27 ) Distributive property 0 = ( w + 18 ) ( w – 18 ) ( w + 27 ) Difference of two squares pattern w + 18 = 0 or w – 18 = 0 w = –18 or w = 18 or w + 27 = 0 Zero-product or w = –27 property Solve for w. Because the width cannot be negative, the only solution is w = 18
    15. 15. Example 6 Solve a multi-step problem STEP 3 Find the length and height. Length w + 36 = 18 + 36 = 54 Height = w – 9 = 18 – 9 = 9 ANSWER The length is 54 inches, the width is 18 inches, and the height is 9 inches.
    16. 16. 9.9 Warm-Up (Day 1) Factor the expression. 1. x(x - 2)+ (x - 2) 2. a + 3a + a + 3 3. y + 2x + yx + 2y 3 2 2
    17. 17. 9.9 Warm-Up (Day 2) Factor the expression. 1. 4y -16y 2. 6g - 24g + 24g 3. h + 4h - 25h -100 6 3 3 4 2 2
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