Polynomial Equations and Factoring
Lessons in this chapter
About Polynomial Equations and Factoring
Polynomial Equations and Factoring teaches several distinct factoring techniques, each suited to a different kind of polynomial.
Solve Polynomial Equations in Factored Form uses the zero product property — if two factors multiply to zero, at least one of them must be zero — to solve an already-factored equation directly.
Factoring Polynomials by Splitting the Middle Term covers the standard technique for factoring a quadratic trinomial into two binomials, while Factoring Polynomials using Special Products reverses the special product patterns from Polynomial Expressions to factor expressions that match those exact patterns.
Factoring Polynomials Completely closes the chapter by combining every technique, pulling out a common factor first, then applying whichever remaining method fits, until a polynomial can't be factored any further.
This chapter is four lessons long and each one opens as an Algebra 1 worksheet in the page: Solve Polynomial Equations in Factored Form, Factoring Polynomials by Splitting the Middle Term, Factoring Polynomials using Special Products, and Factoring Polynomials Completely. The questions are scored as you answer them, and factoring is one of the places where that matters most, since a factored answer can be checked by multiplying it back out. Free to try, with no login needed for your first 3.
Frequently asked questions
Solve Polynomial Equations in Factored Form introduces the zero product property, that if a product is zero then one of its factors must be zero, which is the reason factoring is worth doing at all. Factoring Polynomials by Splitting the Middle Term handles the standard quadratic trinomial, Factoring Polynomials using Special Products recognizes the difference of two squares and perfect square trinomials on sight, and Factoring Polynomials Completely keeps going until no factor can be broken down further, usually starting with a common factor. The chapter sits in the high school Algebra conceptual category (A).
The zero product property states that if two factors multiply to zero, at least one of them must be zero, so a factored equation like (x - 3)(x + 2) = 0 is solved by setting each factor equal to zero separately.
Find two numbers that multiply to give the product of the first and last coefficients but add to give the middle coefficient, then use those two numbers to split the middle term into two terms and factor by grouping.
Recognize the pattern: a difference of two squares, or a perfect square trinomial, and apply the special product formula in reverse to write the factored form directly, without needing to split any middle term.
It means continuing to factor an expression until no factor can be broken down any further, often starting with a greatest common factor, then applying whichever remaining technique fits the leftover expression.
Stopping after the first factoring step (like pulling out a common factor) without checking whether what's left can be factored further, factoring completely requires exhausting every possible technique, not just the first one that applies.