9.9
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9.9 Factor Polynomials Completely
Transcript of 9.9
9.9 Factor Polynomials Completely
Example 1 Factor out a common binomial
Factor the expression.
( )4+x2x ( )4+x3–a.
b. ( )2y3y2 ( )y25 –– +
Distributive property
( )32x – ( )4+x=
SOLUTION
( )4+x2x ( )4+x3–a.
The binomials y 2 and 2 y are opposites. from 2 y to obtain y 2 as a common binomial factor.
b. – –––
–Factor 1
Example 1 Factor out a common binomial
( )53y2 ( )–– y 2= Distributive property
( )2y3y2 ( )y25 –– + ( )2y3y2 ( )5 –– – y 2= Factor 1 from (2 y).
––
Factor by GroupingIn a polynomial with 4 terms, factor a
common monomial from pairs of terms, then look for a common binomial factor.
Example 2 Factor by grouping
Factor 3x2x3 3.x++ +
3x2x3 3x++ + = Group terms. ( )3x2x3 + ( )x 3++
( )1x2 + ( )x 3+= Distributive property
( )3xx2 ( )1 x 3++ += Factor each group; write x 3as 1(x 3).
++
Example 3 Factor by grouping
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.
–
Factor 6x3 3x22x+– –
6x3 3x22x+– – Rearrange terms.x3 2x+– – 63x2=
= ( )3x2x3 ( )2x 6+– – Group terms.
( )3xx2 ( )2 x 3+= – – Factor each group.
( )2x2 += ( )x 3– Distributive property
Example 3 Factor by grouping
CHECK
Check your factorization using a graphing calculator. Graph and . Because the graphs coincide, you know that your factorization is correct.
6x3 3x22x+– –=y1 ( )2x2 += ( )x 3–y2
Factoring CompletelyA factorable polynomial with integer
coefficients is factored completely if it is written as a product of unfactorable polynomials with integer coefficients.
Guidelines for Factoring Polynomials Completely
1. Factor out the greatest common monomial factor. (9.5)
2. Look for a difference of two squares or a perfect square trinomial. (9.8)
3. Factor a trinomial of the form into a product of binomial factors. (9.6 & 9.7)
4. Factor a polynomial with four terms by grouping. (9.9)
ANSWER The correct answer is B.
Multiple Choice PracticeExample 4
Which is the completely factored form of 10n12n2 –+ 8?
( )1+2n( )43n2 – ( )12n –( )43n2 +
( )4n +( )26n3 –( )22n +( )23n3 –
( )43n2 ( )12n –+= Factor trinomial.
SOLUTION
10n12n2 –+ 8 ( )5n 4+6n22 –= Factor out 2.
Example 5 Solve a polynomial equation
Solve 18x23x3 + – 24x.=
SOLUTION
18x23x3 + – 24x= Write original equation.
18x23x3 + 0=24x+ Add 24x to each side.
( )8+x23x 6x + 0= Factor out 3x.
( )2x3x ( )4x + 0=+ Factor trinomial.
0=3x 0=2x + 0=4x +or or Zero-product property
4=x –0=x or or Solve for x.=x 2–
Example 5 Solve a polynomial equation
ANSWER
The solutions of the equation are 0, 2, and 4.– –
Check each solution by substituting it for x in the equation. One check is shown here.
CHECK
( 2– )33 ( 2– )218+ =?
24– ( 2– )
48 48=
7224– 48=+?
Example 6
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.
TERRARIUM
Solve a multi-step problem
Example 6 Solve a multi-step problem
STEP 2 Solve the equation for w.
( )36+w8748 ( )w ( )9w –= Write equation.
w3 27w20 = + – 324w – 8748 Multiply. Subtract 8748 from each side.
SOLUTION
STEP 1 Write a verbal model. Then write an equation.
8748 ww 36+( )= • • w 9( )–
Example 6 Solve a multi-step problem
( )27w2+w3= ( )8748+324w–0 Group terms.
( )27+ww2 ( )27+w324–0 = Factor each group.
0 = ( )324w2 – ( )27w + Distributive property
0 = ( )18w + ( )27w +( )18w – Difference of two squares pattern
018w =–018w =+ 027w =+or or Zero-product property
–18w = –27w =18w =or or Solve for w.
Because the width cannot be negative, the only solution is 18w =
Example 6 Solve a multi-step problem
STEP 3 Find the length and height.
=36w + 3618 + 54=Length
Height 9w –= = 918 – = 9
ANSWER
The length is 54 inches, the width is 18 inches, and the height is 9 inches.
9.9 Warm-Up (Day 1)Factor the expression.
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9.9 Warm-Up (Day 2)Factor the expression.
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