Factor Theorem

High School

Definition

Helps to show the connection between the factors and zeroes of a polynomial. The factor theorem tells us that f(x) has a factor of (x - k) if f(k) = 0, meaning that k is a root of the polynomial.

Worked examples

\(f(x) = x^3 - 6x^2 + 11x - 6\); test \(x = 2\): \(f(2) = 8 - 24 + 22 - 6 = 0\), so \((x - 2)\) is a factor.

Because \(f(2) = 0\), the Factor Theorem guarantees \((x - 2)\) divides the polynomial.

\(f(x) = x^2 + 5x + 6\); \(f(-2) = 4 - 10 + 6 = 0\) and \(f(-3) = 9 - 15 + 6 = 0\), so factors are \((x + 2)(x + 3)\).

Both \(x = -2\) and \(x = -3\) are roots, so both \((x + 2)\) and \((x + 3)\) are factors.

Common mistakes

\(f(2) = 0\) means \((x + 2)\) is a factor → \(f(2) = 0\) means \((x - 2)\) is a factor If \(k\) is a root, the factor is \((x - k)\), not \((x + k)\). Watch the sign.
Testing \(f(3) = 5\) (non-zero) proves \((x - 3)\) is not a factor, so 3 is not a root → Correct — \(f(3) \ne 0\) means \((x - 3)\) is not a factor and 3 is not a root This is actually correct logic; the mistake is thinking a non-zero result means anything else.
If \((x - 4)\) is a factor, then \(f(4)\) equals 4 → If \((x - 4)\) is a factor, then \(f(4) = 0\) A factor means the root makes the polynomial zero, not equal to the root itself.

Where you'll use it next

You'll use the Factor Theorem to find all factors and roots of polynomials, solve polynomial equations, perform synthetic and long division, and sketch polynomial graphs in precalculus and calculus.

Found in 1 StudyPug lesson

Factor Theorem: The Cornerstone of Polynomial Mathematics

Grade 12 Math

Dive into the Factor Theorem and revolutionize your approach to polynomials. Master factoring, root-finding, and graphing techniques. Elevate your algebra prowess and conquer complex equations with confidence.

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