Factor trinomials'

1
Section 5.4

• Factor trinomials.

2
Factoring Trinomials

• Factoring trinomials involves a certain amount of
  trial and error.
• Seeing patterns in factored forms can reduce the
  amount of trial and error that is needed.

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Factoring Trinomials
4
Factoring Trinomials
5
Factoring Trinomials
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7
Factor x2 10x 9

• Step 1 x2 10x 9 (x )(x )

8
Factor x2 10x 9

• Step 1 x2 10x 9 (x )(x )
• Step 2 List pairs of factors of the constant
  9
• 9 33 and 9 91

9
Factor x2 10x 9

• Step 1 x2 10x 9 (x )(x )
• Step 2 List pairs of factors of the constant
  9
• 9 33 and 9 91
• Step 3 Try various combinations of factors
• (x 3)(x 3) x2 3x 3x 9 x2 9

10
Factor x2 10x 9

• Step 1 x2 10x 9 (x )(x )
• Step 2 List pairs of factors of the constant
  9
• 9 33 and 9 91
• Step 3 Try various combinations of factors
• (x 3)(x 3) x2 3x 3x 9 x2 9
• (x 3)(x 3) x2 3x 3x 9 x2 6x 9

11
Factor x2 10x 9

• Step 1 x2 10x 9 (x )(x )
• Step 2 List pairs of factors of the constant
  9
• 9 33 and 9 91
• Step 3 Try various combinations of factors
• (x 3)(x 3) x2 3x 3x 9 x2 9
• (x 3)(x 3) x2 3x 3x 9 x2 6x 9
• (x 3)(x 3) x2 3x 3x 9 x2 6x 9

12
Factor x2 10x 9

• Step 1 x2 10x 9 (x )(x )
• Step 2 List pairs of factors of the constant
  9
• 9 33 and 9 91
• Step 3 Try various combinations of factors
• (x 3)(x 3) x2 3x 3x 9 x2 9
• (x 3)(x 3) x2 3x 3x 9 x2 6x 9
• (x 3)(x 3) x2 3x 3x 9 x2 6x 9
• (x 9)(x 1) x2 x 9x 9 x2 10x 9

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Factor x2 10x 9

• Step 1 x2 10x 9 (x )(x )
• Step 2 List pairs of factors of the constant
  9
• 9 33 and 9 91
• Step 3 Try various combinations of factors
• (x 3)(x 3) x2 3x 3x 9 x2 9
• (x 3)(x 3) x2 3x 3x 9 x2 6x 9
• (x 3)(x 3) x2 3x 3x 9 x2 6x 9
• (x 9)(x 1) x2 x 9x 9 x2 10x 9
• So x2 10x 9 (x 9)(x 1)

14
Factor a2 a 12

• Step 1 a2 a 12 (a )(a )

15
Factor a2 a 12

• Step 1 a2 a 12 (a )(a )
• Step 2 List pairs of factors of the constant
  9
• 12 34 and 12 62 and 12 112

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Factor a2 a 12

• Step 1 a2 a 12 (a )(a )
• Step 2 List pairs of factors of the constant
  9
• 12 34 and 12 62 and 12 112
• Step 3 Try various combinations of factors
• (a 4)(a 3) a2 3a 4a 12 a2 7a 12

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Factor a2 a 12

• Step 1 a2 a 12 (a )(a )
• Step 2 List pairs of factors of the constant
  9
• 12 34 and 12 62 and 12 112
• Step 3 Try various combinations of factors
• (a 4)(a 3) a2 3a 4a 12 a2 7a 12
• (a 4)(a 3) a2 3a 4a 12 a2 7a 12

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Factor a2 a 12

• Step 1 a2 a 12 (a )(a )
• Step 2 List pairs of factors of the constant
  9
• 12 34 and 12 62 and 12 112
• Step 3 Try various combinations of factors
• (a 4)(a 3) a2 3a 4a 12 a2 7a 12
• (a 4)(a 3) a2 3a 4a 12 a2 7a 12
• (a 4)(a 3) a2 3a 4a 12 a2 a 12

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Factor a2 a 12

• Step 1 a2 a 12 (a )(a )
• Step 2 List pairs of factors of the constant
  9
• 12 34 and 12 62 and 12 112
• Step 3 Try various combinations of factors
• (a 4)(a 3) a2 3a 4a 12 a2 7a 12
• (a 4)(a 3) a2 3a 4a 12 a2 7a 12
• (a 4)(a 3) a2 3a 4a 12 a2 a 12
  
• (a 4)(a 3) a2 3a 4a 12 a2 a 12

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Factor a2 a 12

• Step 1 a2 a 12 (a )(a )
• Step 2 List pairs of factors of the constant
  9
• 12 34 and 12 62 and 12 112
• Step 3 Try various combinations of factors
• (a 4)(a 3) a2 3a 4a 12 a2 7a 12
• (a 4)(a 3) a2 3a 4a 12 a2 7a 12
• (a 4)(a 3) a2 3a 4a 12 a2 a 12
  
• (a 4)(a 3) a2 3a 4a 12 a2 a 12
• So a2 a 12 (a 4)(a 3)

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Tip

• To factor x2 bx c when c is positive, find
  two numbers with the same sign as the middle
  term.
• x2 5x 5 (x 3)(x 2)
• x2 14x 24 (x 12)(x 2)

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Tip

• To factor x2 bx c when c is negative, find
  two numbers with opposite signs whose sum is the
  coefficient of the middle term.
• x2 7x 60 (x 12)(x 5)

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2 variables

• x2 9xy 14y2

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2 variables

• x2 9xy 14y2

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2 variables

• x2 9xy 14y2
• x2 9xy 14y2 (x ?y)(x ?y)

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2 variables

• x2 9xy 14y2
• x2 9xy 14y2 (x ?y)(x ?y)
• c is positive so we use the same sign,
  specifically 2 negatives. (x ?y)(x ?y)
• What divisors of 14 sum to 9?

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2 variables

• x2 9xy 14y2
• x2 9xy 14y2 (x ?y)(x ?y)
• c is positive so we use the same sign,
  specifically 2 negatives. (x ?y)(x ?y)
• What divisors of 14 sum to 9?
• 72 14 and 7 2 9

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2 variables

• x2 9xy 14y2
• x2 9xy 14y2 (x ?y)(x ?y)
• c is positive so we use the same sign,
  specifically 2 negatives. (x ?y)(x ?y)
• What divisors of 14 sum to 9?
• 72 14 and 7 2 9
• x2 9xy 14y2 (x 2y)(x 7y)

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• 3x2 3x 18

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• 3x2 3x 18
• 3x2 3x 18 3(x2 x 6)

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• 3x2 3x 18
• 3x2 3x 18 3(x2 x 6)
• 3(x2 x 6) 3(x ?)(x ?)

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• 3x2 3x 18
• 3x2 3x 18 3(x2 x 6)
• 3(x2 x 6) 3(x ?)(x ?)
• c is negative so we use the opposite signs, whose
  sum is 1. 3(x ?)(x ?)

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• 3x2 3x 18
• 3x2 3x 18 3(x2 x 6)
• 3(x2 x 6) 3(x ?)(x ?)
• c is negative so we use the opposite signs, whose
  sum is 1. 3(x ?)(x ?)
• What divisors of 6 sum to 1?

34

• 3x2 3x 18
• 3x2 3x 18 3(x2 x 6)
• 3(x2 x 6) 3(x ?)(x ?)
• c is negative so we use the opposite signs, whose
  sum is 1. 3(x ?)(x ?)
• What divisors of 6 sum to 1?
• 3(-2) -6 and 3 -2 1

35

• 3x2 3x 18
• 3x2 3x 18 3(x2 x 6)
• 3(x2 x 6) 3(x ?)(x ?)
• c is negative so we use the opposite signs, whose
  sum is 1. 3(x ?)(x ?)
• What divisors of 6 sum to 1?
• 3(-2) -6 and 3 -2 1
• 3x2 3x 18 3(x 2)(x 3)

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Factor 2x2 9x 7

• 2x2 9x 7 ( )( )
• 2x2 (2x)(x) and 7 (7)(1)
• So 2x2 9x 7 (2x )(x )
• c is positive so use same signs
• Try various combinations of factors to finish
• (2x 1)(x 7) 2x2 14x x 7 2x2 15x
  7
• (2x 7)(x 1) 2x2 2x 7x 7 2x2 9x
  7
• So 2x2 9x 7 (2x 7)(x 1)

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Factor 2x2 9x 7

• 2x2 9x 7 ( )( )

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Factor 2x2 9x 7

• 2x2 9x 7 ( )( )
• 2x2 (2x)(x) and 7 (7)(1)
• So 2x2 9x 7 (2x )(x )

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Factor 2x2 9x 7

• 2x2 9x 7 ( )( )
• 2x2 (2x)(x) and 7 (7)(1)
• So 2x2 9x 7 (2x )(x )
• c is positive so use same signs
• Try various combinations of factors to finish

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Factor 2x2 9x 7

• 2x2 9x 7 ( )( )
• 2x2 (2x)(x) and 7 (7)(1)
• So 2x2 9x 7 (2x )(x )
• c is positive so use same signs
• Try various combinations of factors to finish
• (2x 1)(x 7) 2x2 14x x 7 2x2 15x
  7

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Factor 2x2 9x 7

• 2x2 9x 7 ( )( )
• 2x2 (2x)(x) and 7 (7)(1)
• So 2x2 9x 7 (2x )(x )
• c is positive so use same signs
• Try various combinations of factors to finish
• (2x 1)(x 7) 2x2 14x x 7 2x2 15x
  7
• (2x 7)(x 1) 2x2 2x 7x 7 2x2 9x
  7
• So 2x2 9x 7 (2x 7)(x 1)

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Factor Trinomials by Grouping

• Factoring ax2 bx c by grouping
• Multiply a times c.
• Find factors of ac whose sum is b.
• Rewrite bx as the sum/difference of the factors
  found above.
• Factor by grouping

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Factor 6x2 19x 15

• 6x2 19x 15

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Factor 6x2 19x 15

• 6x2 19x 15
• (6)(15) 90 (10)(9) and 9 10 19

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Factor 6x2 19x 15

• 6x2 19x 15
• (6)(15) 90 (10)(9) and 9 10 19
• 6x2 19x 15 6x2 10x 9x 15

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Factor 6x2 19x 15

• 6x2 19x 15
• (6)(15) 90 (10)(9) and 9 10 19
• 6x2 19x 15 6x2 10x 9x 15
• 2x(3x 5) 3(3x 5)

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Factor 6x2 19x 15

• 6x2 19x 15
• (6)(15) 90 (10)(9) and 9 10 19
• 6x2 19x 15 6x2 10x 9x 15
• 2x(3x 5) 3(3x 5)
• (3x 5)(2x 3)

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Factor 4x2 27x 18

• 4x2 27x 18

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Factor 4x2 27x 18

• 4x2 27x 18
• (4)(18) 72 (24)(3) and 24 3 27

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Factor 4x2 27x 18

• 4x2 27x 18
• (4)(18) 72 (24)(3) and 24 3 27
• 4x2 27x 18 4x2 24x 3x 18

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Factor 4x2 27x 18

• 4x2 27x 18
• (4)(18) 72 (24)(3) and 24 3 27
• 4x2 27x 18 4x2 24x 3x 18
• 4x(x 6) 3(x 6)

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Factor 4x2 27x 18

• 4x2 27x 18
• (4)(18) 72 (24)(3) and 24 3 27
• 4x2 27x 18 4x2 24x 3x 18
• 4x(x 6) 3(x 6)
• (x 6)(4x 3)