7 Factoring Ax2+Bx+c

8/13/2019 7 Factoring Ax2+Bx+c

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8 Factoring ax2+ bx+ c

Warm Up

Lesson Presentation

California Standards

Preview


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8 Factoring ax2+ bx+ cWarm Up

Find each product.1. (x2)(2x + 7)

2. (3y+ 4)(2y + 9)

3. (3n5)(n

7)

Factor each trinomial.

4. x2+ 4x 32

5.z2 + 15z+ 36

6. h217h+ 72

6y2+ 35y + 36

2x2+ 3x 14

3n226n+ 35

(z+ 3)(z+ 12)

(x4)(x+ 8)

(h8)(h9)


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11.0 Students apply basic factoring techniquesto second-and simple third- degreepolynomials. These techniques include finding acommon factor for all terms in a polynomial,recognizing the difference of two squares, and

recognizing perfect squares of binomials.

California

Standards


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When you multiply (3x+ 2)(2x+ 5), the coefficientof thex2-term is the product of the coefficients ofthex-terms. Also, the constant term in the trinomial

is the product of the constants in the binomials.

(3x+2)(2x+5) = 6x2

+ 19x+10


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8 Factoring ax2+ bx+ cAdditional Example 1: Factoring ax2+ bx+ c

Factor 6x2

+ 11x

+ 4. Check your answer.

The first term is 6x2, so at least

one variable term has a

coefficient other than 1.

( x + )( x+ )

The coefficient of thex2term is 6. The constant termin the trinomial is 4.

Try integer factors of

6 for the

coefficients and

integer factors of

4 for the constant

terms.

(1x+ 4)(6x + 1) = 6x2+ 25x+ 4(1x+ 2)(6x + 2) = 6x2+ 14x+ 4(1x+ 1)(6x + 4) = 6x2+ 10x+ 4(2x+ 4)(3x + 1) = 6x2+ 14x+ 4

(3x+ 4)(2x + 1) = 6x2+ 11x+ 4


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8 Factoring ax2+ bx+ cPartner Share!Example 1a

Factor each trinomial. Check your answer.The first term is 6x2, so at least



( x + )( x+ )

The coefficient of thex2term is 6. The constant term inthe trinomial is 3.

6x2+ 11x+ 3

(1x+ 3)(6x + 1) = 6x2+ 19x+ 3(1x+ 1)(6x + 3) = 6x2+ 9x+ 3

Try integer factors of

6 for the

coefficients andinteger factors of 3

for the constant

terms.

(2x+ 1)(3x + 3) = 6x2+ 9x+ 3

(3x+ 1)(2x + 3) = 6x2+ 11x+ 3


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8 Factoring ax2+ bx+ cPartner Share!Example 1b

Factor each trinomial. Check your answer.The first term is 3x2, so at least



( x + )( x+ )

3x22x8

The coefficient of thex2term is 3. The constant term inthe trinomial is 8.

Try integer factors of 3

for the coefficients

and integer factorsof 8 for the constant

terms.

(1x1)(3x + 8) = 3x2+ 5x8(1x8)(3x + 1) = 3x223x8(1x4)(3x + 2) = 3x

2

10x

8

(1x2)(3x + 4) = 3x2 2x8


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So, to factor ax2+ bx+ c, check the factors of aandthe factors of cin the binomials. The sum of theproducts of the outer and inner terms should be b.

( X+ )( x+ ) =ax2+bx+c

Sum of outer and inner products = b

Product = cProduct = a


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Since you need to check all the factors of aand allthe factors of c, it may be helpful to make atable. Then check the products of the outer andinner terms to see if the sum is b. You can multiplythe binomials to check your answer.

( X+ )( x+ ) =ax2+bx+c

Sum of outer and inner products = b

Product = cProduct = a


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8 Factoring ax2+ bx+ cAdditionalExample 2A: Factoring ax2+ bx+ c

When cis Positive

Factor each trinomial. Check your answer.2x2+ 17x+ 21

( x+ )( x+ )a = 2 and c = 21; Outer + Inner = 17.

(x+ 7)(2x+ 3)

Factors of 2 Factors of 21 Outer+Inner

1and 2 1and 21 1(21)+ 2(1)= 23

1and 2 21and 1 1(1)+ 2(21)= 43

1and 2 3and 7 1(7)+ 2(3)= 13

1and 2 7and 3 1(3)+ 2(7)= 17

Check (x+ 7)(2x+ 3)= 2x2+ 3x+ 14x+ 21

= 2x2+ 17x+ 21Use the FOILmethod.


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When bis negative and cis positive, the factorsof care both negative.

Remember!


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Factor each trinomial. Check your answer.3x216x+ 16

a = 3 and c = 16,

Outer + Inner = 16 .

(x4)(3x

4)

Check (x4)(3x4) = 3x24x12x + 16

= 3x216x + 16 Use the FOIL method.


1and 3 1and 16 1(16)+ 3(1)=19

1and 3 2 and 8 1( 8) + 3(2)=14

1and 3 4and 4 1( 4)+ 3(4)= 16

( x+ )( x+ )

AdditionalExample 2B: Factoring ax2+ bx+ c

When cis Positive


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Factor each trinomial. Check your answer.6x2+ 17x+ 5 a = 6 and c = 5;

Outer + Inner = 17.


1and 6 1and 5 1(5)+ 6(1)= 11

2and 3 1and 5 2(5)+ 3(1)= 13

3and 2 1and 5 3(5)+ 2(1)= 17

(3x+ 1)(2x+ 5)Check(3x+ 1)(2x+ 5)= 6x2+ 15x+ 2x+ 5

= 6x2+ 17x+ 5Use the FOIL method.

( x+ )( x+ )


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Factor each trinomial. Check your answer.9x215x+ 4 a = 9 and c = 4; Outer + Inner =

15.


3and 3 1and 4 3(4)+ 3(1)=15

3and 3 2 and 2 3(2) + 3(2)=12

3and 3 4and 1 3(1)+ 3(4)=15

(3x4)(3x

1)

Check (3x4)(3x1) = 9x23x12x + 4

= 9x215x + 4Use the FOIL method.

( x+ )( x+ )


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Factor each trinomial. Check your answer.3x2+ 13x+ 12 a = 3 and c = 12; Outer + Inner

= 13.


1and 3 1and 12 1(12)+ 3(1)= 15

1and 3 2and 6 1(6)+ 3(2)= 12

1and 3 3and 4 1(4)+ 3(3)= 13

(x+ 3)(3x+ 4)

Check (x+ 3)(3x+ 4) = 3x2+ 4x+ 9x + 12

= 3x2+ 13x + 12Use the FOIL method.

Partner Share!Example 2c

( x+ )( x+ )


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When cis negative, one factor of cwill bepositive and the other factor will be negative.

Only some of the factors are shown in theexamples, but you may need to check all of thepossibilities.


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8 Factoring ax2+ bx+ cAdditionalExample 3A: Factoring ax2+ bx+ c

When cis Negative

Factor each trinomial. Check your answer.3n2+ 11n4

( n+ )( n+ )

a = 3 and c = 4;

Outer + Inner = 11.

(n+ 4)(3n1)

Check (n+ 4)(3n1) = 3n2 n + 12n 4= 3n2+ 11n 4

Use the FOIL method.


1and 31and 4 1(4)+ 3(1)= 1

1and 3 2 and 2 1(2) + 3(2)=4

1and 3 4and 1 1(1)+ 3(4) = 11

1and 3 4and 1 1(1)+ 3(4) = 11


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Factor each trinomial. Check your answer.

2x2+ 9x18

( x+ )( x+ )

a = 2 and c = 18;

Outer + Inner = 9 .


1and 2 18 and

1 1(

1)+ 2(18)= 351and 2 9 and2 1(2) + 2(9)= 16

1and 2 6 and3 1(3)+ 2(6) = 9

(x+ 6)(2x3)

Check (x+ 6)(2x3) = 2x23x+ 12x 18

= 2x2+ 9x 18Use the FOIL method.

AdditionalExample 3B: Factoring ax2+ bx+ c

When cis Negative


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2


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Factor each trinomial. Check your answer.6x

2

+ 7x

3( x+ )( x+ )

a = 6 and c =

3; Outer + Inner= 7.


6and 1 1 and3 6(3)+ 1(1)=17

6and 1 3 and

1 6(

1) + 1(3)=

3

3and 2 3(3)+ 2(1)=7

3and 2 3(1) + 2(3)= 3 1 and33 and12and 3 2(3)+ 3(1)=3

2and 3 1(2) + 3(3)= 7

1 and3

3 and1

(2x+ 3)(3x 1)

Check(2x+ 3)(3x 1) = 6x22x+ 9x 3

Use the FOIL method.

= 6x+ 7x 3

2


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Factor each trinomial. Check your answer.4n2n3

( n+ )( n+ )

a = 4 and c = 3;

Outer + Inner = 1.

(n1)(4n + 3) Use the FOIL method.

Factors of 4 Factors of

3 Outer+Inner1and 4 1 and 3 1(3)+1(4)= 1

1and 4 1 and 3 1(3) 1(4)=1

Check (n1)(4n + 3) = 4n2+ 3n 4n 3

= 4n2n 3

F t i 2 b


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When the leading coefficient is negative,factor out 1 from each term before usingother factoring methods.


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F t i 2 b


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8 Factoring ax2+ bx+ cAdditionalExample 4: Factoring ax2+ bx +c

When ais Negative

Factor

2x2

5x

3.

1(2x2+ 5x+ 3)

1( x+ )( x+ )

Factor out 1.

a = 2 and c = 3;

Outer + Inner = 5


1and 2 3 and1 1(1) + 3(2)= 71and 2 1 and 3 1(3)+1(2)= 5

1(x+ 1)(2x + 3)

(x+ 1)(2x + 3)

F t i 2 b


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Factor each trinomial. Check your answer.6x217x121(6x2+ 17x+ 12)

1( x+ )( x+ )

Factor out 1.

a = 6 and c = 12;

Outer + Inner = 17Factors of 6 Factors of 12 Outer+Inner

2and 3 4 and3 2(3) +3(4)= 182and 3 3 and 4 2(4)+3(3)= 17

(2x+ 3)(3x + 4)1(2x+ 3)(3x + 4)

Check 1(2x + 3)(3x+ 4) = 6x28x 9x 12= 6x217x 12

F t i 2 + b +


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Factor each trinomial. Check your answer.3x217x101(3x2+ 17x+ 10)

1( x+ )( x+ )

Factor out 1.

a = 3 and c = 10;

Outer + Inner = 17)


1and 3 2 and5 1(5) +3(2)= 111and 3 5 and 2 1(2)+3(5)= 17

(x + 5)(3x+ 2)1(x + 5)(3x+ 2)

Check 1(x+ 5)(3x + 2) = 3x22x 15x 10

= 3x217x 10

F t i 2 + b +


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8 Factoring ax2+ bx+ cLesson Review!

Factor each trinomial. Check your answer.

1.5x2+ 17x+ 6

2.2x2+ 5x 12

3.6x223x+ 7

4.4x2+ 11x+ 20

5. 2x2+ 7x3

6.8x2+ 27x + 9

(x+ 4)(4x+ 5)

(3x1)(2x

7)

(2x 3)(x + 4)

(5x + 2)(x+ 3)

(2x+ 1)(x3)

(8x+ 3)(x+ 3)

7 Factoring Ax2+Bx+c

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