Factoring quadratic polynomials
Factoring a quadratic is like un-doing the “FOIL” process
Factoring of quadratic polynomials (second-degree polynomials) is done by “un-FOILing,” which means we start with the result of a FOIL problem and work backwards to find the two binomial factors.
Hi! I'm krista.
I create online courses to help you rock your math class. Read more.
To factor a quadratic polynomial in which the term has a coefficient of and the constant term is nonzero (in other words, a quadratic polynomial of the form where ), you’ll be considering pairs of factors of the last term (the constant term) and finding the pair of factors whose sum is the coefficient of the middle term (the -term).
How to factor quadratics
Take the course
Want to learn more about Algebra 1? I have a step-by-step course for that. :)
Looking for the factors of a constant
Example
Factor the quadratic polynomial.
Start by listing the pairs of factors of the constant term, , and their sums. We’re looking for the pair of factors whose sum is (the coefficient of the -term).
Since and have a sum of , they’re the factors we need. The answer is
To check our answer, we can FOIL .
Let’s try another example of factoring a quadratic polynomial.
Factoring of quadratic polynomials (second-degree polynomials) is done by “un-FOILing,” which means we start with the result of a FOIL problem and work backwards to find the two binomial factors.
Example
Factor the quadratic polynomial.
Start by listing the pairs of factors of and their sums. We’re looking for the pair of factors whose sum is (the coefficient of the -term).
The factors and have a sum of , so they’re the correct factors.
To check our answer, we can FOIL .
If the coefficient of the term in a quadratic polynomial is either or the greatest common factor of the polynomial, we can first factor that out and then use the procedure described above to factor what’s left over.
Example
Factor the quadratic polynomial.
The greatest common factor of this polynomial is , so we first factor out a .
Since and , we see that can be factored as follows:
So the given quadratic polynomial can be factored as
In later lessons, you’ll learn how to factor more complicated quadratic polynomials - those in which all of the following conditions are satisfied:
The coefficient of the term is neither nor .
The coefficient of the term isn’t the greatest common factor of the polynomial.
The constant term is nonzero.