8-4 Factoring ax 2 + bx + c Warm Up Warm Up Lesson Presentation Lesson Presentation California...
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Transcript of 8-4 Factoring ax 2 + bx + c Warm Up Warm Up Lesson Presentation Lesson Presentation California...
8-4 Factoring ax2 + bx + c
Warm Up
Find each product.
1. (x – 2)(2x + 7)
2. (3y + 4)(2y + 9)
3. (3n – 5)(n – 7)
Find each trinomial.4. x2 + 4x – 325. z2 + 15z + 366. h2 – 17h + 72
6y2 + 35y + 36
2x2 + 3x – 14
3n2 – 26n + 35
(z + 3)(z + 12) (x – 4)(x + 8)
(h – 8)(h – 9)
8-4 Factoring ax2 + bx + c
11.0 Students apply basic factoring techniques to second- and simple third- degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.
California Standards
8-4 Factoring ax2 + bx + c
In the previous lesson you factored trinomials of the form x2 + bx + c. Now you will factor trinomials of the form ax2 + bx + c, where a ≠ 0 or 1.
8-4 Factoring ax2 + bx + c
When you multiply (3x + 2)(2x + 5), the coefficient of the x2-term is the product of the coefficients of the x-terms. Also, the constant term in the trinomial is the product of the constants in the binomials.
(3x + 2)(2x + 5) = 6x2 + 19x + 10
8-4 Factoring ax2 + bx + c
To factor a trinomial like ax2 + bx + c into its binomial factors, first write two sets of parentheses: ( x + )( x + ).
Write two integers that are factors of a next to the x’s and two integers that are factors of c in the other blanks. Then multiply to see if the product is the original trinomial. If there are no two such integers, we say the trinomial is not factorable.
8-4 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 the x2 term is 6. The constant term in 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
8-4 Factoring ax2 + bx + cAdditional Example 1 Continued
Factor 6x2 + 11x + 4. Check your answer.
6x2 + 11x + 4 = (3x + 4)(2x + 1)
The factors of 6x2 + 11x + 4 are (3x + 4) and (2x + 1).
Check (3x + 4)(2x + 1) = Use the FOIL
method.6x2 + 3x + 8x + 4
= 6x2 + 11x + 4The product of the original
trinomial.
8-4 Factoring ax2 + bx + cCheck It Out! Example 1a
Factor each trinomial. 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 the x2 term is 6. The constant term in the 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 and integer factors of 3 for the constant terms.
(2x + 1)(3x + 3) = 6x2 + 9x + 3
(3x + 1)(2x + 3) = 6x2 + 11x + 3
8-4 Factoring ax2 + bx + cCheck It Out! Example 1a Continued
Factor each trinomial. Check your answer.
The factors of 6x2 + 11x + 3 are (3x + 1)(2x + 3).
6x2 + 11x + 3 = (3x + 1)(2x +3)
Check (3x + 1)(2x + 3) = Use the FOIL method.
6x2 + 9x + 2x + 3
= 6x2 + 11x + 3
The product of the original trinomial.
8-4 Factoring ax2 + bx + cCheck It Out! Example 1b
Factor each trinomial. Check your answer.
The first term is 3x2, so at least one variable term has a coefficient other than 1.
( x + )( x + )
3x2 – 2x – 8
The coefficient of the x2 term is 3. The constant term in the trinomial is –8.
Try integer factors of 3 for the coefficients and integer factors of 8 for the constant terms.
(1x – 1)(3x + 8) = 3x2 + 5x – 8
(1x – 8)(3x + 1) = 3x2 – 23x – 8 (1x – 4)(3x + 2) = 3x2 – 10x – 8
(1x – 2)(3x + 4) = 3x2 – 2x – 8
8-4 Factoring ax2 + bx + cCheck It Out! Example 1b
Factor each trinomial. Check your answer.3x2 – 2x – 8
The factors of 3x2 – 2x – 8 are (x – 2)(3x + 4).
3x2 – 2x – 8 = (x – 2)(3x + 4)
Check (x – 2)(3x + 4) = Use the FOIL method.3x2 + 4x – 6x – 8
= 3x2 – 2x – 8 The product of the original trinomial.
8-4 Factoring ax2 + bx + c
So, to factor ax2 + bx + c, check the factors of a and the factors of c in the binomials. The sum of the products of the outer and inner terms should be b.
( X + )( x + ) = ax2 + bx + c
Sum of outer and inner products = b
Product = cProduct = a
8-4 Factoring ax2 + bx + c
Since you need to check all the factors of a and all the factors of c, it may be helpful to make a table. Then check the products of the outer and inner terms to see if the sum is b. You can multiply the binomials to check your answer.
( X + )( x + ) = ax2 + bx + c
Sum of outer and inner products = b
Product = cProduct = a
8-4 Factoring ax2 + bx + cAdditional Example 2A: Factoring ax2 + bx + c When c
is PositiveFactor 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
1 and 2 1 and 21 1(21) + 2(1) = 23
1 and 2 21 and 1 1(1) + 2(21) = 43 1 and 2 3 and 7 1(7) + 2(3) = 13 1 and 2 7 and 3 1(3) + 2(7) = 17
Check (x + 7)(2x + 3) = 2x2 + 3x + 14x + 21= 2x2 + 17x + 21
Use the FOILmethod.
8-4 Factoring ax2 + bx + c
When b is negative and c is positive, the factors of c are both negative.
Remember!
8-4 Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
3x2 – 16x + 16a = 3 and c = 16, Outer + Inner = –16 .
(x – 4)(3x – 4)
Check (x – 4)(3x – 4) = 3x2 – 4x – 12x + 16
= 3x2 – 16x + 16
Use the FOIL method.
Factors of 3 Factors of 16 Outer + Inner
1 and 3 –1 and –16 1(–16) + 3(–1) = –19 1 and 3 – 2 and – 8 1( – 8) + 3(–2) = –14 1 and 3 – 4 and – 4 1( – 4) + 3(– 4)= –16
( x + )( x + )
Additional Example 2B: Factoring ax2 + bx + c When c is Positive
8-4 Factoring ax2 + bx + cCheck It Out! Example 2a
Factor each trinomial. Check your answer.
6x2 + 17x + 5 a = 6 and c = 5; Outer + Inner = 17.
Factors of 6 Factors of 5 Outer + Inner
1 and 6 1 and 5 1(5) + 6(1) = 11
2 and 3 1 and 5 2(5) + 3(1) = 13 3 and 2 1 and 5 3(5) + 2(1) = 17
(3x + 1)(2x + 5)Check (3x + 1)(2x + 5) = 6x2 + 15x + 2x + 5
= 6x2 + 17x + 5
Use the FOIL method.
( x + )( x + )
8-4 Factoring ax2 + bx + cCheck It Out! Example 2b
Factor each trinomial. Check your answer.
9x2 – 15x + 4 a = 9 and c = 4; Outer + Inner = –15.
Factors of 9 Factors of 4 Outer + Inner
3 and 3 –1 and – 4 3(–4) + 3(–1) = –15 3 and 3 – 2 and – 2 3(–2) + 3(–2) = –12 3 and 3 – 4 and – 1 3(–1) + 3(–4)= –15
(3x – 4)(3x – 1)
Check (3x – 4)(3x – 1) = 9x2 – 3x – 12x + 4
= 9x2 – 15x + 4
Use the FOIL method.
( x + )( x + )
8-4 Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
3x2 + 13x + 12 a = 3 and c = 12; Outer + Inner = 13.
Factors of 3 Factors of 12 Outer + Inner
1 and 3 1 and 12 1(12) + 3(1) = 15 1 and 3 2 and 6 1(6) + 3(2) = 12 1 and 3 3 and 4 1(4) + 3(3) = 13
(x + 3)(3x + 4)
Check (x + 3)(3x + 4) = 3x2 + 4x + 9x + 12
= 3x2 + 13x + 12
Use the FOIL method.
Check It Out! Example 2c
( x + )( x + )
8-4 Factoring ax2 + bx + c
When c is negative, one factor of c will be positive and the other factor will be negative. Only some of the factors are shown in the examples, but you may need to check all of the possibilities.
8-4 Factoring ax2 + bx + cAdditional Example 3A: Factoring ax2 + bx + c When c
is NegativeFactor each trinomial. Check your answer.
3n2 + 11n – 4
( n + )( n+ )
a = 3 and c = – 4; Outer + Inner = 11.
(n + 4)(3n – 1)Check (n + 4)(3n – 1) = 3n2 – n + 12n – 4
= 3n2 + 11n – 4
Use the FOIL method.
Factors of 3 Factors of –4 Outer + Inner
1 and 3 –1 and 4 1(4) + 3(–1) = 1 1 and 3 –2 and 2 1(2) + 3(–2) = – 4 1 and 3 –4 and 1 1(1) + 3(–4) = –11
1 and 3 4 and –1 1(–1) + 3(4) = 11
8-4 Factoring ax2 + bx + c
Factor each trinomial. Check your answer.2x2 + 9x – 18
( x + )( x+ )a = 2 and c = –18;
Outer + Inner = 9 .
Factors of 2 Factors of –18 Outer + Inner
1 and 2 18 and –1 1(–1) + 2(18) = 35 1 and 2 9 and –2 1(–2) + 2(9) = 16 1 and 2 6 and –3 1(–3) + 2(6) = 9
(x + 6)(2x – 3)
Check (x + 6)(2x – 3) = 2x2 – 3x + 12x – 18
= 2x2 + 9x – 18
Use the FOIL method.
Additional Example 3B: Factoring ax2 + bx + c When c is Negative
8-4 Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
4x2 – 15x – 4
( x + )( x+ )a = 4 and c = –4;
Outer + Inner = –15.
Factors of 4 Factors of – 4 Outer + Inner
1 and 4 –1 and 4 1(4) – 1(4) = 0 1 and 4 –2 and 2 1(2) – 2(4) = –6 1 and 4 –4 and 1 1(1) – 4(4) = –15
(x – 4)(4x + 1) Use the FOIL method.
Check (x – 4)(4x + 1) = 4x2 + x – 16x – 4
= 4x2 – 15x – 4
Additional Example 3C: Factoring ax2 + bx + c When c is Negative
8-4 Factoring ax2 + bx + cCheck It Out! Example 3a
Factor each trinomial. Check your answer.
6x2 + 7x – 3
( x + )( x+ )a = 6 and c = –3; Outer + Inner
= 7.
Factors of 6 Factors of –3 Outer + Inner
6 and 1 1 and –3 6(–3) + 1(1) = –17 6 and 1 3 and –1 6(–1) + 1(3) = – 3
3 and 2 3(–3) + 2(1) = –7 3 and 2 3(–1) + 2(3) = 3
1 and –3 3 and –1
2 and 3 2(–3) + 3(1) = –3 2 and 3 1(–2) + 3(3) = 7
1 and –3 3 and –1
(2x + 3)(3x – 1)Check (2x + 3)(3x – 1) = 6x2 – 2x + 9x – 3
Use the FOIL method.
= 6x + 7x – 3
8-4 Factoring ax2 + bx + cCheck It Out! Example 3b
Factor each trinomial. Check your answer.
4n2 – n – 3
( n + )( n+ )
a = 4 and c = –3; Outer + Inner = –1.
(n – 1) (4n + 3) Use the FOIL method.
Factors of 4 Factors of –3 Outer + Inner
1 and 4 1 and –3 1(–3) + 1(4) = 1 1 and 4 –1 and 3 1(3) – 1(4) = –1
Check (n – 1)(4n + 3) = 4n2 + 3n – 4n – 3
= 4n2 – n – 3
8-4 Factoring ax2 + bx + c
When the leading coefficient is negative, factor out –1 from each term before using other factoring methods.
8-4 Factoring ax2 + bx + c
When you factor out –1 in an early step, you must carry it through the rest of the steps and into the answer.
Caution!
8-4 Factoring ax2 + bx + cAdditional Example 4: Factoring ax2 + bx + c When
a is NegativeFactor –2x2 – 5x – 3.
–1(2x2 + 5x + 3)
–1( x + )( x+ )
Factor out –1. a = 2 and c = 3;
Outer + Inner = 5Factors of 2 Factors of 3 Outer + Inner
1 and 2 3 and 1 1(1) + 3(2) = 7 1 and 2 1 and 3 1(3) + 1(2) = 5
–1(x + 1)(2x + 3)
(x + 1)(2x + 3)
8-4 Factoring ax2 + bx + cCheck It Out! Example 4a
Factor each trinomial. Check your answer.
–6x2 – 17x – 12 –1(6x2 + 17x + 12)
–1( x + )( x+ )
Factor out –1.
a = 6 and c = 12; Outer + Inner = 17
Factors of 6 Factors of 12 Outer + Inner
2 and 3 4 and 3 2(3) + 3(4) = 18 2 and 3 3 and 4 2(4) + 3(3) = 17
(2x + 3)(3x + 4) –1(2x + 3)(3x + 4) Check –1(2x + 3)(3x + 4) = –6x2 – 8x – 9x – 12
= –6x2 – 17x – 12
8-4 Factoring ax2 + bx + cCheck It Out! Example 4b
Factor each trinomial. Check your answer.
–3x2 – 17x – 10 –1(3x2 + 17x + 10)
–1( x + )( x+ )
Factor out –1. a = 3 and c = 10; Outer + Inner = 17)
Factors of 3 Factors of 10 Outer + Inner
1 and 3 2 and 5 1(5) + 3(2) = 11 1 and 3 5 and 2 1(2) + 3(5) = 17
(x + 5)(3x + 2)–1(x + 5)(3x + 2)Check –1(x + 5)(3x + 2) = –3x2 – 2x – 15x – 10
= –3x2 – 17x – 10