Factor Theorem
In short: The factor theorem connects zeros and factors: a binomial of the form x – c divides a polynomial with no leftover precisely when plugging c into the polynomial gives 0.
Algebra 2AlgebraA-APR.B.2
What is Factor Theorem?
The factor theorem tells you whether a binomial goes into a polynomial evenly, and it settles the question without dividing: just evaluate the polynomial at a single number.
Take any polynomial \(f(x)\). The binomial \(x – c\) is one of its factors exactly when \(f(c) = 0\). So every zero \(c\) produces a linear factor \(x – c\), and every linear factor reveals a zero. The link runs both ways, which makes it a favorite tool for breaking cubics and higher-degree polynomials into pieces.
How to spot it
- You are given a polynomial and a possible factor such as x – 4 and must decide whether it works.
- One zero of a cubic is provided and the problem asks you to factor the whole polynomial.
- A question says that a certain value makes the polynomial equal to zero and asks what factor follows.
- You need an unknown coefficient so that a stated binomial divides the polynomial evenly.
The method
- Take the candidate binomial \(x – c\) and find \(c\); for \(x + 3\) use \(c = -3\).
- Evaluate \(f(c)\) by substituting \(c\) for \(x\).
- If \(f(c) = 0\), then \(x – c\) is a factor; if not, it is not a factor.
- To finish factoring, divide \(f(x)\) by \(x – c\) and keep factoring the quotient.
Factor Theorem: solved example
Problem
Show that \(x – 4\) is a factor of \(g(x) = x^3 – 6x^2 + 5x + 12\), then factor the cubic completely.
Solution
Here \(c = 4\). Compute \(g(4) = 64 – 96 + 20 + 12 = 0\). The value is zero, so \(x – 4\) is a factor.
Divide to get the quotient \(x^2 – 2x – 3\), which factors as \((x – 3)(x + 1)\). Therefore \(g(x) = \)\((x – 4)(x – 3)(x + 1)\).
Try it
Question 1
Is \(x + 2\) a factor of \(x^3 + 3x^2 – 4\)?
Show the answer
\(c = -2\): \(-8 + 12 – 4 = 0\). Yes, \(x + 2\) is a factor.
Question 2
Find \(k\) so that \(x – 1\) is a factor of \(x^2 + kx – 10\).
Show the answer
Need \(f(1) = 0\): \(1 + k – 10 = 0\), so \(k = 9\).
Factor Theorem vs. Remainder Theorem
The remainder theorem tells you what is left after dividing by \(x – c\), and that leftover is \(f(c)\) for any \(c\). The factor theorem asks only about the case \(f(c) = 0\) and concludes that \(x – c\) divides the polynomial exactly. One gives a number, the other gives a yes-or-no verdict.
Keep going
Lessons
Worksheets and quizzes
- Dividing Polynomials and the Remainder Theorem – Worksheet
- Dividing Polynomials and the Remainder Theorem – Quiz
- Zeros and Graphs of Polynomial Functions – Worksheet
- Zeros and Graphs of Polynomial Functions – Quiz
Related words
- Remainder Theorem
- Zeros of a Function
- Polynomial Long Division
- Synthetic Division
- Rational Root Theorem
- Fundamental Theorem of Algebra
Whole dictionary: A–Z · Algebra 2 · Algebra