5'3: Factoring Polynomials

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5.3 Factoring Polynomials

• Review Multiplication of polynomials
• (-xy)(4x3y5)
• -4x4y6
• (2x - 5)8x3
• 16x4 - 40x3
• (x 3)(x 4) use distributive property
• X(x4) 3(x4)multiply / combine like terms
• x2 7x 12

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More Review?

• (3x - 1)(2x 4)
• 3x(2x4) -1(2x 4)
• (2x - 5)(2x 5)
• (x 4)2
• 2x(x-3)(x2)

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Factoring out common factors

• 4x2 5x
• Any common factors?
• Only the factor x is present in both terms
• Factoring out the x we get ..
• x(4x 5)
• 12x3 - 4x2
• 4x3y2 x2y3

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Solving an equation by factoring

• Zero-product Property
• If (a)(b) 0, then either a 0, or b 0, or
  both a b 0
• We will use this property to solve certain
  equations by factoring
• 2x2 4x 0
• Factoring the left side of the eqn (lse) we get..
• 2x(x 2) 0

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Solve using the zero product property

• 2x(x 2) 0
• Either 2x 0 or x 2 0 or both
• If 2x 0, x 0 if x 2 0, x -2
• We have 2 possible solutions 0, -2
• 4x2 3x 0
• 0, -3/4
• 32x4 - 16x3
• 0, 1/2

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Solve using the zero product property

• 34x2 51x
• Re-write the eqn
• 34x2 - 51x 0 . Factor and solve
• 0, 3/2
• 5x3 6x2 - 4x3
• Move all terms to lse, simplify, and solve
• 0, 2/3

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Factoring by Grouping

• A technique that makes use of the associative and
  distributive properties.
• Factor 2x(x 1) 3(x 1)
• Solution Since both terms in the expression
  contain the binomial (x 1), the distributive
  property can be used to factor the expression.
• 2x(x 1) 3(x 1) (2x 3)(x 1)

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Factoring by Grouping

• X3 - 2x2 3x - 6
• Out of our league, unless
• Use Associative Property
• (x3 - 2x2) (3x - 6)
• Factor out GCF
• X2(x - 2) 3(x - 2)
• Re-write using Distributive Property
• (x2 3)(x - 2) (see slide 1, 2 return 7)

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Factoring by Grouping

• 4x3 3x2 8x 6
• (4x3 3x2) (8x 6)
• X2(4x 3) 2(4x 3)
• (x2 2)(4x 3)
• Check your answer in your TI-83
• Enter the problem in Y1
• Enter your answer in Y2
• Check table values for Y1, Y2 Same?

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Solving Eqns on the TI-83

• 15x - 5x2 0
• Enter lse in Y1 rse in Y2
• Adjust window so you can see the graph
• Since the graphs cross in 2 places, well need to
  calculate 2 solutions
• When Guess?, well need to move the cursor
  close to the intersection point were looking for.

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Solving Eqns on the TI-83

• h(t) -16t2 128t
• Re-write the eqn -16x2 128x 0
• Enter lse and rse in Y1/Y2
• Adjust window as necessary, keeping in mind that
  the lse is a parabola . Well need to see it
  cross the x-axis in 2 places
• Calculate the intersection points

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Homework 5.3

• 11 - 20 25 - 30 35 - 38 43 - 46 55 - 58 77 -
  80 93.