What Is a Polynomial?

A polynomial is an expression made up of terms involving non-negative integer powers of a variable. The general form is:

P(x) = a_nxⁿ + a_n−1xⁿ⁻¹ + … + a₁x + a₀

where n is a non-negative integer and a_n ≠ 0

Degree 1 (Linear)

P(x) = 2x + 3

Degree 2 (Quadratic)

P(x) = x² − 4x + 4

Degree 3 (Cubic)

P(x) = x³ − 2x² + x

The Remainder Theorem

When a polynomial P(x) is divided by (x − a), the remainder equals P(a). This means you can find the remainder without performing long division.

Remainder Theorem

If P(x) ÷ (x − a), remainder = P(a)

1

Worked Example: Find the remainder

Find the remainder when P(x) = x³ − 3x + 2 is divided by (x − 2).

Step 1: By the remainder theorem, remainder = P(2)

Step 2: P(2) = (2)³ − 3(2) + 2 = 8 − 6 + 2 = 4

The remainder is 4.

The Factor Theorem

The factor theorem is a special case of the remainder theorem. If P(a) = 0, then (x − a) is a factor of P(x). Conversely, if (x − a) is a factor, then P(a) = 0.

Factor Theorem

(x − a) is a factor of P(x) ⇔ P(a) = 0

2

Worked Example: Factorise a cubic

Fully factorise P(x) = x³ − 6x² + 11x − 6.

Step 1: Test x = 1: P(1) = 1 − 6 + 11 − 6 = 0. So (x − 1) is a factor.

Step 2: Divide: P(x) = (x − 1)(x² − 5x + 6)

Step 3: Factorise the quadratic: x² − 5x + 6 = (x − 2)(x − 3)

Answer: P(x) = (x − 1)(x − 2)(x − 3)

3

Worked Example: Sketching a cubic

Sketch P(x) = (x − 1)(x − 2)(x − 3).

x-intercepts: x = 1, x = 2, x = 3 (where each factor equals zero)

y-intercept: P(0) = (−1)(−2)(−3) = −6

End behaviour: Leading term is x³, so rises to +∞ as x → +∞, falls to −∞ as x → −∞

Shape: Cubic “S-curve” crossing x-axis at all three roots

Key Vocabulary

Degree

The highest power of the variable in a polynomial. A cubic has degree 3.

Root / Zero

A value of x where P(x) = 0. Roots correspond to x-intercepts on the graph.

Leading Coefficient

The coefficient of the highest-degree term. Determines the end behaviour of the graph.

Monic Polynomial

A polynomial whose leading coefficient equals 1, e.g. x³ − 5x + 2.

Knowledge Check

Apply the remainder and factor theorems, and test your polynomial knowledge.

Question 1

What is the degree of the polynomial P(x) = 4x³ − 2x² + 7?

2 3 4 7

Question 2

Using the remainder theorem, find the remainder when P(x) = x³ + 2x − 5 is divided by (x − 1).

0 5 −2 −2

Question 3

If P(3) = 0, which of the following is a factor of P(x)?

(x + 3) (x² − 9) (3x) (x − 3)

Question 4

How many x-intercepts can a cubic polynomial have at most?

3 2 1 Unlimited

Question 5

Which value of k makes (x − 2) a factor of P(x) = x² + kx − 6?

k = 1 k = −1 k = 2 k = −3

Polynomials