Sec' 5'4: Factoring Trinomials

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Sec. 5.4 Factoring Trinomials

• Factor trinomials in the form x2 bx c
• Factor trinomials in the form ax2 bx c
• Factor trinomials using technology

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Factoring Trinomials

• Reverse FOIL
• Requires thought
• Requires trial and error
• Requires an algorithm a precise method.

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X2 8x 12

• Smile! The leading coefficient is 1! Easy!
• Set up for FOIL
• ( )( )
• We know First term has to be x because only xx
  x2
• We know last term has to be factors of 12 12,1
  or 4,3 or 6,2.
• (x 12)(x 1) or (x 4)(x 3) or (x 6)(x 2)

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X2 8x 12

• The sign before the 12 lets us know we will
  be adding the two factors.
• The sum of the 2 factors must 8!
• Of the 3 pairs of factors only 6 and 2 have a sum
  of 8
• The sign before the 12 also lets us know both
  signs in the solution will be the same.

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X2 8x 12

• Possibilities for solution
• (x - 6)(x - 2) or
• (x 6)(x 2)
• (-6)(-2) 12 and (6)(2) 12
• But -6 -2 -8 6 2 8
• Sooooo.
• (x 6)(x 2) check by FOIL or TI-83

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X2 - 7x 12

• Here again our leading coefficient is 1
• (x )(x )
• The last terms must be factors of 12 6,2 or
  12,1 or 3,4.
• The before the 12 tells us we will be adding
  the 2 factors, and that the signs will be the
  same!
• The sum of the factors must be -7!

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X2 - 7x 12

• Of the 3 pairs of factors of 12, only 4 3 sum
  to 7
• Signs must be the same, so
• (4)(3) 12 4 3 7
• (-4)(-3) 12 -4 -3 -7 these are the
  factors we are looking for!
• (x - 4)(x - 3)
• Check by FOIL or TI-83

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X2 - 4x - 12

• Leading coefficient is 1 we need two factors of
  12 whose difference is 4
• The - sign in front of the 12 also tells us
  that the signs will be different in our solution.
• Factors of 12 whose difference is 4 6 2
• The - before the 4 lets us know the sign of the
  larger number (6) must be negative
• (x - 6)(x 2)

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Practice!

• X2 13x - 30
• X2 x - 30
• X2 6x - 16
• X2 11x 30
• X2 - 15x - 16

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Solve x2 6x - 16 using TI-83

• Set up an equation X2 6x - 16 0
• Enter lse in Y1 and rse in Y2
• Adjust window to see graphs intersect
• Calc point of intersection
• X -8 x 2
• Set each eqn 0
• X 8 0 x - 2 0 . And by 0 product
  property
• (x 8)(x - 2) 0

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Factoring ax2 bx c

• More difficult!
• Requires more trial error, or
• Box or Table method
• Factor by grouping

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Factoring ax2 bx c

• I dont recommend using trial error .. Most
  students dont have the patience to keep going
  until theyve found the solution.
• The Box method and Factoring by grouping are
  essentially the same. The box just keeps the
  terms nice and neat.

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2x2 9x 4 using Box

• First term last term in opposite corners
• Diagonal products are equal
• 2 Factors of 8x2 whose sum is 9x
• Factor our GCF of each row
• Check

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2x2 9x 4 using Trial Error

• Perhaps by now we can recognize that both signs
  in the factors will be
• In this case we only have 3 possibilites
• (2x 4)(x 1)
• (2x 1)(x 4)
• (2x 2)(x 2)
• Check by FOIL (or TI-83) to see which is a
  solution.

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4x3 - 14x2 6x

• Factor out GCF
• 2x(2x2 - 7x 3)
• Factor 2x2 - 7x 3 using Box or TE

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Practice

• 2x2 - 3x - 9
• 3x2 5x - 2
• 3x2 - 9x - 4

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Lets try -100x2 10,000x - 240,000 0

• Can we divide through by anything to make the
  numbers smaller?
• It would also be nice to have our x2 term
  positive instead of negative. How?
• X2 - 100x 2400 0

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Factoring with Technology

• Factor x2 - 3x - 4 using the table below

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5.4 Homework

• 8-9 21-26 33-38 71-72 77-79 84-85 93.