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Factoring polynomials

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  • 1. Factoring Polynomials
    NCVPS
    Summer 2010
  • 2. Factoring is reversing the process of multiplying monomials, binomials or trinomials or any combination of these.
    We will focus on factoring out monomials (we call these GCFs) from a polynomial or factoring trinomials into two binomials.
    Ex: 3x(a – 3) or (x – 1)(x + 5)
    | | /
    monomial binomial binomials
    What is factoring?
  • 3. Sometimes a common factor is present.
    Ex 1: 3x2 – 6x -9 has a common factor of 3:
    3(x2 – 2x – 3)
    Ex 2: x3-5x2+4x has a common x in all 3 terms:
    x(x2 - 5x + 4)
    Ex 3: 2x2a + 6xa+12a has a common factor of 2a:
    2a(x2 + 3x + 6)
    Step 1: Look for a GCF present
  • 4. Factor first term into its products.
    Example 1: x2– 2x – 3.
    The first term is x2. It is factored as x*x.
    Factor the last term into its products.
    The last term is -3. It factors as 1(-3)
    or (-1)(3)
    Step 2: Factor first & last terms
  • 5. Example 1: x2 – 2x – 3.
    (x )(x )
    Next, add the factors of -3:
    x2– 2x – 3
    (x -3)(x +1)
    Example 1 continued
  • 6. It is very important to check your factoring to make sure you got the signs in the correct place. Multiply them out again!
    x2 - 2x – 3 = (x -3)(x +1)
    F: x(x)
    O: (-3)(1) = -3
    I: -3x
    L: 1x
    Inner + Last terms = -2x
    Check the work!
  • 7. Checking the signs is very important!
    x2 + 3x + 2 has all + signs. It factors with + signs: (x + 2)(x + 1)
    x2 – 6x + 5 has a +5 but the middle term is -. So 5 must factor as (-5)(-1): (x - 5)(x – 1)
    More examples on the next page 
    Signs, signs, everywhere there’s signs!
  • 8. X2 -2x – 8 has a -8 that factors as + and –
    Here we work with factors and sums. We want a sum of the factors to be -2 (the middle term)
    (-8)(1) = -8 sum the factors: -8+1 = -7
    (8)(-1) = -8 sum the factors: 8 + (-1) = 7
    (-4)(2) = -8 sum the factors: -4 + 2 = -2
    This last one is the factorization we want!
    (x – 4)(x + 2)
    Keep on checking signs!
  • 9. Find the factorization of x2 – 4x -12
    -12 factors as (-3)(4) or (3)(-4) or (2)(-6) or (-2)(6) or
    (-1)(12) or (1)(-12).
    Which of these sums to -4?
    -3 + 4 = 1
    3 + (-4) = -1
    2 + (-6) = -4 We have a winner!
    x2 -4x – 12 is (x + 2)(x – 6)
    Another example
  • 10. x2 – 2x – 15
    2x2 – x – 3
    3x2-12x +9
    Go to the next slide for the solutions.
    You Try Problems
  • 11. x2 – 2x – 15 = (x – 5)(x + 3)
    2x2 – x – 3 = (2x - 3)(x +1)
    3x2-12x +9 = 3(x2 – 4x + 3)
    = 3(x – 3)(x -1)
    You Try Problems