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Factoring polynomials

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This document discusses factoring polynomials. It explains that factoring is reversing the process of multiplying monomials, binomials, or trinomials. The document focuses on factoring out the greatest common factor (GCF) from a polynomial and factoring trinomials into two binomials. It provides examples of factoring polynomials by finding the GCF, factoring the first and last terms of a trinomial, and checking that the signs are correct when factoring. It emphasizes that the signs must be placed correctly when factoring polynomials.

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Factoring PolynomialsNCVPS Summer 2010
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 ¨C 3) or (x ¨C 1)(x + 5) | | / monomial binomial binomialsWhat is factoring?
Sometimes a common factor is present.Ex 1: 3x2 ¨C 6x -9 has a common factor of 3:3(x2 ¨C 2x ¨C 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
Factor first term into its products.Example 1: x2 ¨C 2x ¨C 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
Example 1: x2 ¨C 2x ¨C 3. (x )(x ) Next, add the factors of -3: x2 ¨C 2x ¨C 3 (x -3)(x +1)Example 1 continued
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 ¨C 3 = (x -3)(x +1)F: x(x)O: (-3)(1) = -3I: -3xL: 1x Inner + Last terms = -2x Check the work!
Checking the signs is very important!x2 + 3x + 2 has all + signs. It factors with + signs: (x + 2)(x + 1)x2 ¨C 6x + 5 has a +5 but the middle term is -. So 5 must factor as (-5)(-1): (x - 5)(x ¨C 1)More examples on the next page ?Signs, signs, everywhere there¡¯s signs!
X2 -2x ¨C 8 has a -8 that factors as + and ¨CHere 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 = -2This last one is the factorization we want! (x ¨C 4)(x + 2)Keep on checking signs!
Find the factorization of x2 ¨C 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 = 13 + (-4) = -12 + (-6) = -4 We have a winner!x2 -4x ¨C 12 is (x + 2)(x ¨C 6)Another example
x2 ¨C 2x ¨C 152x2 ¨C x ¨C 33x2-12x +9Go to the next slide for the solutions.You Try Problems
x2 ¨C 2x ¨C 15 = (x ¨C 5)(x + 3)2x2 ¨C x ¨C 3 = (2x - 3)(x +1)3x2-12x +9 = 3(x2 ¨C 4x + 3) = 3(x ¨C 3)(x -1)You Try Problems

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Factoring polynomials

  • 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 ¨C 3) or (x ¨C 1)(x + 5) | | / monomial binomial binomialsWhat is factoring?
  • 3. Sometimes a common factor is present.Ex 1: 3x2 ¨C 6x -9 has a common factor of 3:3(x2 ¨C 2x ¨C 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 ¨C 2x ¨C 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 ¨C 2x ¨C 3. (x )(x ) Next, add the factors of -3: x2 ¨C 2x ¨C 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 ¨C 3 = (x -3)(x +1)F: x(x)O: (-3)(1) = -3I: -3xL: 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 ¨C 6x + 5 has a +5 but the middle term is -. So 5 must factor as (-5)(-1): (x - 5)(x ¨C 1)More examples on the next page ?Signs, signs, everywhere there¡¯s signs!
  • 8. X2 -2x ¨C 8 has a -8 that factors as + and ¨CHere 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 = -2This last one is the factorization we want! (x ¨C 4)(x + 2)Keep on checking signs!
  • 9. Find the factorization of x2 ¨C 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 = 13 + (-4) = -12 + (-6) = -4 We have a winner!x2 -4x ¨C 12 is (x + 2)(x ¨C 6)Another example
  • 10. x2 ¨C 2x ¨C 152x2 ¨C x ¨C 33x2-12x +9Go to the next slide for the solutions.You Try Problems
  • 11. x2 ¨C 2x ¨C 15 = (x ¨C 5)(x + 3)2x2 ¨C x ¨C 3 = (2x - 3)(x +1)3x2-12x +9 = 3(x2 ¨C 4x + 3) = 3(x ¨C 3)(x -1)You Try Problems