Factor Theorem: Explained with Examples, Notes and FAQs

The factor theorem states that (x - a) is a factor of a polynomial p(x) if and only if p(a) = 0. In other words, a linear expression divides a polynomial exactly when the polynomial evaluates to zero at that value. It is the remainder theorem with the remainder set to zero.

This matters because it converts factorisation, which feels like guessing, into testing: you try candidate values, and each zero you find hands you a guaranteed factor. Combined with polynomial division, it cracks cubics and higher-degree polynomials that no other school method can touch.

Key Concepts

• Factor: an expression that divides the polynomial with zero remainder.
• Zero of a polynomial: a value a with p(a) = 0. Zeros and linear factors come in pairs: zero a means factor (x - a).
• "If and only if": the statement works both ways. If (x - a) is a factor then p(a) = 0, and if p(a) = 0 then (x - a) is a factor.
• Candidate roots: for integer-coefficient polynomials, test the factors of the constant term first, with both positive and negative signs.

How It Works

1. List candidate values: the factors of the constant term, each with a plus and minus sign.
2. Substitute each into p(x) until you find a value a with p(a) = 0.
3. Then (x - a) is a factor. Divide the polynomial by (x - a) to reduce its degree.
4. Repeat on the resulting lower-degree polynomial until it is fully factorised.

Worked Example

Factorise x cubed + 2x squared - x - 2 completely.

1. Candidates from the constant term -2: plus or minus 1, 2.
2. Test x = -1: (-1) + 2(1) - (-1) - 2 = -1 + 2 + 1 - 2 = 0. So (x + 1) is a factor.
3. Divide: x cubed + 2x squared - x - 2 = (x + 1)(x squared + x - 2).
4. Factorise the quadratic: x squared + x - 2 = (x + 2)(x - 1).
5. Full factorisation: (x + 1)(x + 2)(x - 1).

Verify by expanding (x + 1)(x + 2) = x squared + 3x + 2, then multiplying by (x - 1): x cubed + 3x squared + 2x - x squared - 3x - 2 = x cubed + 2x squared - x - 2. It matches.

Common Misconceptions

• If p(-2) = 0, the factor is (x - 2). Wrong. The factor is (x - (-2)) = (x + 2). The sign inside the bracket is the opposite of the root.
• One zero gives the complete factorisation. Wrong. One zero gives one factor; you must divide it out and keep going to find the rest.
• The theorem can prove something is NOT a factor. Actually it can: if p(a) is not zero, then (x - a) is definitely not a factor. The test works in both directions.

Real-World Applications

• Solving cubic and quartic equations in physics and engineering starts with the factor theorem to peel off linear factors one by one.
• In control theory, checking whether a system polynomial has a particular root uses exactly this substitution test.
• Graphing software uses these ideas to locate x-intercepts of polynomial curves accurately.

To recap: p(a) = 0 means (x - a) is a factor, and vice versa. Test factors of the constant term, mind the sign when writing the bracket, divide out each factor you find, and keep going until the polynomial is fully broken down.