Factoring Polynomials Using the Factor Theorem
Simple Explanation
To fully factor a higher-degree polynomial: test candidate values of a (usually factors of the constant term) using the factor theorem until P(a) = 0 is found, divide out that factor (by synthetic division), and repeat on the lower-degree quotient until it is fully factored.
Why Do We Need It?
This combines everything from the chapter into the complete practical method for factoring — and, by extension, solving — any polynomial equation of degree 3 or higher.
Formula
The Factor Theorem
(x − a) is a factor of P(x) ⇔ P(a) = 0
A linear expression (x − a) divides a polynomial P(x) exactly (with zero remainder) exactly when a is a root of P(x) — substituting a into P(x) gives zero.
- P(x)
- — the polynomial being tested
- a
- — a candidate root — the constant in the linear factor (x − a)
When to use it: Whenever you need to test whether a specific value is a root of a polynomial, or to find factors of a polynomial by testing candidate values.
Worked Example
Fully factor a cubic polynomial
Fully factor P(x) = x³ − 2x² − 5x + 6.
Why Does This Work?
Each application of the factor theorem removes exactly one linear factor and reduces the polynomial's degree by one — repeating this process (test, divide, repeat) on the shrinking quotient eventually reaches a quadratic (or lower), which ordinary factoring methods can finish.
Real-Life Example
Finding all break-even points of a cubic cost model
A business's profit is modeled by a cubic polynomial, and the owner wants every value where profit is exactly zero.
Fully factoring the cubic using the factor theorem reveals every root — every break-even point — at once, rather than solving a cubic equation from scratch.
Practice
Fully factor x³ − 7x + 6. (Hint: try a = 1 first.)
HardCommon mistake
Stopping after finding just one factor — a cubic generally has up to three roots; keep applying the process to the shrinking quotient until it is fully factored (down to linear or irreducible quadratic factors).
Quick Review
- Test candidate values (factors of the constant term) with the factor theorem.
- Divide out each factor found (synthetic division), then repeat on the smaller quotient.
- Continue until the quotient is a quadratic (or lower) that ordinary methods can finish.