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
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Example 1: x2 – 2x – 3. (x )(x ) Next, add the factors of -3: x2– 2x – 3 (x -3)(x +1) Example 1 continued
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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
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x2 – 2x – 15 2x2 – x – 3 3x2-12x +9 Go to the next slide for the solutions. You Try Problems
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