Factorising Quadratic Trinomials With Leading Coefficient Other Than 1 Summary

Key Concepts

• Factor Trinomials of the Form \(a{x}^{2}+bx+c\) using Trial and Error:
  1. Write the trinomial in descending order of degrees.
  2. Find all the factor pairs of the first term.
  3. Find all the factor pairs of the third term.
  4. Test all the possible combinations of the factors until the correct product is found.
  5. Check by multiplying.
• Factor Trinomials of the Form \(a{x}^{2}+bx+c\) Using the “ac” Method:
  1. Factor any GCF.
  2. Find the product ac.
  3. Find two numbers m and n that:
     \(\begin{array}{cccc}\text{Multiply to}\phantom{\rule{0.2em}{0ex}}ac\hfill & & & m·n=a·c\hfill \\ \text{Add to}\phantom{\rule{0.2em}{0ex}}b\hfill & & & m+n=b\hfill \end{array}\)
  4. Split the middle term using m and n:
     
  5. Factor by grouping.
  6. Check by multiplying the factors.
• Choose a strategy to factor polynomials completely (updated):
  1. Is there a greatest common factor? Factor it.
  2. Is the polynomial a binomial, trinomial, or are there more than three terms?
     If it is a binomial, right now we have no method to factor it.
     If it is a trinomial of the form \({x}^{2}+bx+c\)
     Undo FOIL \(\left(x\phantom{\rule{1em}{0ex}}\right)\left(x\phantom{\rule{1em}{0ex}}\right)\).
     If it is a trinomial of the form \(a{x}^{2}+bx+c\)
     Use Trial and Error or the “ac” method.
     If it has more than three terms
     Use the grouping method.
  3. Check by multiplying the factors.

Glossary

prime polynomials

Polynomials that cannot be factored are prime polynomials.