Definition: The factor theorem says x – c is a factor of a polynomial f(x) exactly when f(c) equals zero, so roots and linear factors correspond.
What is the statement?
For a polynomial f, x – c is a factor if and only if f(c) = 0. In other words, every root gives a linear factor and every linear factor gives a root. It is the remainder theorem in the special case where the remainder is zero.
How does it help you factor?
Test small candidate roots, then divide to lower the degree.
Worked example: factor f(x) = x3 – 7x + 6. Try x = 1: 1 – 7 + 6 = 0, so x – 1 is a factor. Synthetic division with c = 1 on 1, 0, -7, 6 gives 1, 1, -6, 0, so f(x) = (x – 1)(x2 + x – 6) = (x – 1)(x + 3)(x – 2).
The roots are 1, -3 and 2.
What is a common error?
Reading the root with the wrong sign. If f(-3) = 0, the factor is x + 3, not x – 3. Mathbot’s habit: root c, factor x – c.
Where to learn it
- Dividing Polynomials: Remainder and Factor Theorems – lesson
- Polynomial Functions: Zeros, Ends, and Graphs – lesson