5'5: Special Types of Factoring

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5.5 Special Types of Factoring

• Difference of Squares
• Perfect Square Trinomials
• Sum/Difference of Cubes
• Solving Polynomial Eqns

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Difference of Squares

• Earlier we learned a shortcut for multiplying
  factors in the form
• (a - b)(a b)
• a2 - b2
• (x - 4)(x 4)
• x2 - 16
• (2a 3)(2a - 3)
• 4a2 - 9

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Now Factor!

• b2 - 16
• 2 terms are being subtracted (difference)
• The first term is a perfect square b times
  itself b2
• The second term is a perfect square 4 times
  itself is 16
• We have a Difference of Squares!

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b2 - 16

• What is being squared to yield b2?
• Square root . b
• What is being squared to yield 16?
• 4
• The factors of b2 - 16 are

• (b - 4)(b 4)

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Factor 9x2 - 64

• Observe
• We have a difference of 2 terms
• The first term is a perfect square
• (3x)2 9x2
• The second term is a perfect square
• (8)2 64
• The factors are

(3x 8)(3x - 8)
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Factor 4x2 - 9y2

• We have a difference of 2 terms
• Both terms are perfect squares
• (2x)2 4x2
• (3y)2 9y2
• (2x - 3y)(2x 3y)

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Factor x4 - y6

• We have a difference of 2 terms
• Both terms are perfect squares
• (x2)2 x4
• (y3)2 y6
• (x2 - y3)(x2 y3)

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Factor

• 4x2 - 25
• 36 - y2
• 81x2 - 100y2
• 16x4 - y4

• (2x - 5)(2x 5)
• (6 - y)(6 y)
• (9x - 10y)(9x 10y)
• (4x2 - y2)(4x2 y2)
• 4x2 - y2 is also a diff of squares
• (2x - y)(2x y)(4x2 y2)

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Perfect Square Trinomials

• A trinomial resulting from
• (a b)2, or
• (a - b)2
• (a b)2 a2 2ab b2
• (a - b)2 a2 - 2ab b2
• Observe The first term (of the trinomial) is a
  perfect square the last term is a perfect
  square the middle term is double the product of
  the square roots of the first and last term.

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Factor x2 6x 9

• Is this a perfect square trinomial?
• X2 is a perfect square xx
• 9 is a perfect square 33
• The middle term is double the product of the
  square roots of x2 and 9 2x3 6x
• Since the middle term is positive the factors
  are
• (x 3)(x 3) or (x 3)2

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Factor 81x2 - 72x 16

• Is this a perfect square trinomial?
• 81x2 is a perfect square (9x)2
• 16 is a perfect square (4)2
• 72x 249x
• Since the middle term is negative the factors
  are

• (9x - 4)(9x - 4) or (9x - 4)2

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Factor

• x2 2x 1
• x2 - 6x 9
• 4x2 20x 25
• 49y2 - 42y 9

• (x 1)2
• (x - 3)2
• (2x 5)2
• (7y - 3)2

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Sum / Difference of Cubes

• a3 b3 (a b)(a2 - ab b2)
• a3 - b3 (a - b)(a2 ab b2)
• Observe We must first identify the cube root of
  both terms.
• We then place this root into the formula.

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Factor x3 8

• a3 b3 (a b)(a2 - ab b2)
• a cube root of x3 x
• b cube root of 8 2
• Now plug a b into formula
• (x 2)(x2 - 2x 4)

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Factor 27x3 - 64y3

• a3 - b3 (a - b)(a2 ab b2)
• a cube root of 27x3 3x
• b cube root of 64y3 4y
• Now plug a b into formula
• (3x - 4y)((3x)2 (3x)(4y) (4y)2)
• (3x - 4y)(9x2 12xy 16y2)

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Factor

• a3 b3 (a b)(a2 - ab b2)
• a3 - b3 (a - b)(a2 ab b2)

• X3 27
• X3 - 64

• (x 3)(x2 - 3x 9)
• (x - 4)(x2 4x 16)

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Solve symbolically 16x4 - 64x3 64x2 0

• Factor out GCF
• 16x2(x2 - 4x 4) 0
• Factor the trinomial a PST?
• x2 - 4x 4 (x - 2)2
• 16x2(x - 2)2 0 apply ZPP

(x - 2)2 0 x - 2 0 (sqrt both sides) x
2
16x2 0 x2 0 x 0
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Solve graphically 16x4 - 64x3 64x2 0

• Enter lse in Y1 rse in Y2
• GRAPH
• Adjust Window if necessary to see where graphs
  intersect
• CALC the Intersection of the 2 graphs using
  method previously discusssed.

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5.5 Homework

• 11 - 16
• 37-38
• 41-44
• 49-53
• 67-70