Polynomials • Topic 2 of 4

Factor Theorem

The factor theorem says (x − a) is a factor of a polynomial P(x) exactly when P(a) = 0. It is the fast lane for factorising cubics and higher polynomials that do not split by inspection: test the divisors of the constant term (with the obvious sign possibilities) until one makes P equal to zero, then divide out that factor and finish with a quadratic. For a polynomial with integer coefficients, any integer root must divide the constant term, so the candidate list is short. CAT also uses the theorem in reverse — "find k so that (x − 2) is a factor" simply means set P(2) = 0 and solve for k. A useful corollary: (x − 1) is a factor when the coefficients sum to 0, and (x + 1) is a factor when the alternating sum is 0. Spotting a root first turns a messy factorisation into one clean division.

✅ Solved examples

1. Show that (x − 2) is a factor of x³ − 3x² + 4x − 4.
P(2) = 8 − 12 + 8 − 4 = 0. Since P(2) = 0, (x − 2) is a factor.
2. Find k if (x + 1) is a factor of 2x³ + 3x² − kx + 2.
P(−1) = −2 + 3 + k + 2 = 0 ⇒ k + 3 = 0 ⇒ k = −3.
3. Factorise x³ − 6x² + 11x − 6 completely.
P(1) = 1 − 6 + 11 − 6 = 0, so (x − 1) is a factor. Dividing gives x² − 5x + 6 = (x − 2)(x − 3). Hence (x − 1)(x − 2)(x − 3).
4. Without dividing, decide whether (x − 1) is a factor of 4x³ − 3x² − 2x + 1.
Sum of coefficients = 4 − 3 − 2 + 1 = 0, so P(1) = 0 and (x − 1) is a factor.

✏️ Practice — try these, take hints as needed

1. Is (x − 3) a factor of x³ − 4x² + x + 6?
Compute P(3).
27 − 36 + 3 + 6.
Check whether it is 0.
Yes, P(3) = 0
2. Find k so that (x − 2) is a factor of x³ + kx² − 4x − 8.
Set P(2) = 0.
8 + 4k − 8 − 8 = 0.
4k = 8.
k = 2
3. Factorise x³ + 2x² − 5x − 6 completely.
Try x = 2: P(2) = 0, so (x − 2) is a factor.
Divide to get x² + 4x + 3.
Factor the quadratic.
(x − 2)(x + 1)(x + 3)
4. For what value of a is (x + 2) a factor of x³ − ax² + 2x + 8?
Set P(−2) = 0.
−8 − 4a − 4 + 8 = 0.
−4a − 4 = 0.
a = −1
5. Is (x + 1) a factor of 3x⁴ − 2x³ + x² − x − 7?
Compute P(−1).
3 + 2 + 1 + 1 − 7.
Check if it equals 0.
Yes, P(−1) = 0

📝 Topic test — 8 questions

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