Factoring%20trinomials:%20x2%20 %20bx%20 %20c

1
Lesson 72

• Factoring trinomials x2 bx c

2
Factoring trinomials

• The polynomial x2 bx c is a trinomial
• Like numbers, some trinomials can be factored
  into 2 linear factors.

3
Factoring when c is positive

• Factor x2 9x 18
• In this trinomial b is 9 and c is 18
• Because b is positive, it must be the sum of 2
  positive numbers that are factors of c
• List all factors of c, which is 18
• 1,18 2,9 3,6
• Which pair adds up to b , which is 9?
• 3,6
• So x2 9x 18 (x3)(x6)

4
practice

• Factor x2 5x 6
• x2 8x 15
• x2 4x 4
• x2 4x 3
• x2 7x 12

5
Factoring when c is positive and b is negative

• Factor x2-5x4
• List factors of 4 1,4 2,2 -1, -4 -2,-2
• Which pair adds up to -5 ?
• -1,-4
• x2 -5x 4 (x-1)(x-4)

6
practice

• Factor x2 8x 15
• x2 -7x 12
• x2 -5x 6
• x2 4x 3
• x2 10x 16

7
Factoring when c is negative

• Factor x2 3x -10
• List factors of -10
• -1,10 -10,1 -2,5 -5,2
• Which pair adds up to 3?
• -2,5
• x2 3x -10 (x-2)(x5)

8
practice

• Factor x2 7x 8
• x2 7x 30
• x2 3x -54
• x2 5x - 14

9
Factoring with 2 variables

• x2 5xy 6y2
• b 5y c 6y2
• list factors of 6y2
• 1,6y2 6,y2 2,3y2 3,2y2 y,6y 2y,3y
• Which pair adds up to 5y
• 2y, 3y
• So x2 5xy 6y2 (x2y)(x3y)

10
factor

• x2 2xy 3y2
• x2 4xy 4y2
• x2 -7xy -18y2

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Rearranging terms before factoring

• -21-4xx2
• Write the trinomial in standard form , then
  factor
• x2 -4x-21
• list factors of -21, that add up to -4
• -1,21 1,-21 -3,7 3,-7
• 3, -7 adds up to -4
• so x2-4x-21 (x3)(x-7)

12
practice

• Factor -24 5x x2
• 3x 10 x2
• -8 x2 7x

13
Evaluating trinomials

• Evaluate x25x-14 and its factors
• for x 3
• x2 5x -14 (x7)(x-2)
• (3)2 5(3)-14 (37)(3-2)
• 9 15-14 (10)(1)
• 10 10

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practice

• Evaluate
• x2 5x 14 for x 2

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Lesson 75

• The pattern used for trinomials is different when
  a is not 1.
• Factor 2x2 7x 5
• 2x2 has factors 2x and x so we know
• (2x )( x )
• Factors of 5 are 1,5 -1,-5
• Because the middle term is positive, we can
  eliminate the -1,-5 pair
• Check each other pair to see if the middle terms
  added equal the middle term in the trinomial
• (2x 1)(x5) or (2x 5)(x1)
• 2x2 10xx5 2x22x5x5
• 2x2 11x 5 2x2 7x 5

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factor

• 3x2 13x12
• 6x2 11x3

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Factoring when b is negative and c is positive

• Factor 6x2 - 11x 3
• factors of 6x2 (x )(6x ) or (2x )(3x )
• Factors of 3 1,3 or -1,-3
• Since the middle term is negative, we can
  eliminate 1,3
• Try (x-1)(6x-3) 6x2-3x-6x36x2-9x3
• (x-3)(6x-1)6x2-x-18x36x2-19x3
• (2x-1)(3x-3)6x2-6x-3x36x2-9x3
• (2x-3)(3x-1)6x2-2x-9x36x2-11x3

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practice

• 4x2 -23x 15
• 10x2-23x12

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Factoring when c is negative

• 4x2 4x-3
• Factors of 4x2 (4x )(x )or(2x )(2x )
• Factors of -3 1, -3 or -1,3
• (4x1)(x-3) 4x2-11x-3
• (4x-3)(x1) 4x2x-3
• (4x-1)(x3) 4x211x-3
• (4x3)(x-1) 4x2-x-3
• (2x1)(2x-3) 4x2-4x-3
• (2x-1)(2x3)4x24x-3

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factor

• 6x28x-8
• 2x2-3x-20
• 3x2-11x-4

21
Factoring with 2 variables

• Factor 2x2-11xy5y2
• Factors of 2x2 (2x )(x )
• Factors of 5y2 5y, y or -5y, -y
• Since the middle sign is negative , we can
  eliminate the 5y,y
• (2x-5y)(x-y) or (2x-y)(x-5y)
• 2x2-2xy-5xy5y2 or 2x2-10xy-xy5y2
• 2x2-7xy5y2 or 2x2-11xy5y2

22
practice

• Factor 2x2-5xy2y2
• 6x211xy4y2

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Rearranging before factoring

• Trinomials must be in standard form before
  factoring

24
Lesson 79

• Factoring trinomials by using the GCF

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Factoring trinomials with positive leading
coefficients

• Always factor the GCF first, then continue
  factoring as in previous lessons
• x45x36x2
• GCF is x2, so
• x2(x25x6)
• x2(x3)(x2)

26
Practice

• Factor
• 4x3-4x2-80x
• a53a4-18a3
• 2m4-16m330m2

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Factoring with negative lead coefficients

• It will always be easier to factor a trinomial
  with a positive lead coefficient, so whenever
  possible, factor out a negative if your trinomial
  has a negative lead coefficient.
• -x2x56 -1(x2-x-56)-1(x-8)(x7)