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How Do You Factor a Polynomial by Guessing and Checking?

Factor 6x2+19x+15 by guessing and checking.

Summary

  1. Since both signs in our polynomial are positive, both sides in the binomials will be positive
  2. Find the factors of the first term, 6
  3. Find the factors of the third term, 15
  4. The factors of 6 will be the coefficients of the first terms of our binomials
  5. The factors of 15 are the coefficients of the second terms of our binomials
  6. FOIL the binomials together to see if we have the right answer

Notes

    1. GCF stands for greatest common factor
    1. If a polynomial is in standard form, then all like terms must be combined
    1. If a polynomial is in standard form, then the degrees must get smaller from left to right
    1. The degree of the first term is 2, and the degree of the second term is 1
    1. We want to pick out the correct template to use for our binomials
    1. The last sign of the trinomial will tell us if the signs of the binomials will be the same or different
    1. The last sign of the trinomial will tell us if the signs of the binomials will be the same or different
    1. The first sign of the trinomial will tell us what kind of signs are in the binomials
    1. We want to find all the factors of 6
    1. We want to find all the factors of 15
    1. Remember, we are using an addition sign in each binomial because there were two addition signs in our trinomial
    1. 6•1 and 2•3 are the factors of 6
    1. 1 and 6 are factors of 6
    1. 1 and 6 become the coefficients of the first terms
    1. 1•15 and 3•5 are the factors of 15
    1. So 1 and 15 become the second terms
    1. 2 and 3 are factors of 6
    2. We could also try 1 and 6 again since they are also factors of 6
    1. 2 and 3 are factors of 6
    2. We could also try 1 and 6 again since they are also factors of 6
    1. 3 and 5 are factors of 15
    2. We could also try 1 and 15 again since they are also factors of 15
    1. When we multiply two binomials we have to FOIL
    1. So 6x2+19x+15 factors into the binomials (2x+3) and (3x+5)
    1. A prime polynomial is just a polynomial that cannot be factored