Further Mathematics · Lesson 4

Remainder and factor theorems

WAECNECO

What the syllabus asks

WAEC: Polynomials: addition, subtraction, multiplication and division of polynomials; remainder and factor theorems; factorisation of cubic and higher polynomials; solving polynomial equations; roots and coefficients; partial fractions (linear factors)

NECO: Polynomials: addition, subtraction, multiplication and division of polynomials; remainder and factor theorems; factorisation of cubic and higher polynomials; solving polynomial equations; roots and coefficients; partial fractions (linear factors)

What you will learn

Add, subtract, multiply and evaluate polynomials.
Divide polynomials by long division and synthetic division.
State and apply the remainder and factor theorems.
Find unknown coefficients from given remainders or factors.
Factorise cubics and solve cubic equations.
Use relationships between roots and coefficients.
Resolve simple rational functions into partial fractions.

Key points

A polynomial has whole-number powers; degree = highest power.
f(x) = d(x)·Q(x) + R(x), with degree R < degree d.
Synthetic division by (x − a): bring down, multiply by a, add; last number is the remainder.
Remainder theorem: R = f(a) for (x − a), f(q/p) for (px − q).
Factor theorem: f(a) = 0 ⟺ (x − a) is a factor.
Factorise cubics: trial root from factors of the constant, divide, factorise the quadratic.
Two unknowns: one equation per condition, solve simultaneously, check.
Division by a quadratic gives a linear remainder px + q.
Quadratic roots: α + β = −b/a, αβ = c/a; cubic: sum −b/a, pairs c/a, product −d/a.
Partial fractions: A/(x − a) + B/(x − b); for repeated factors include A/(x − a) and B/(x − a)².

Start here: What is a polynomial?

A polynomial in x is an expression made of terms of the form axⁿ, where n is a whole number (0, 1, 2, …) and a is a constant. For example f(x) = 3x⁴ − 2x² + x − 7.

Degree = highest power of x. Degree 1: linear; 2: quadratic; 3: cubic; 4: quartic.
A polynomial is written in descending powers of x.

Expressions such as 1/x, √x or 2ˣ are not polynomials, because their powers are negative, fractional, or the variable is in the power.

Evaluating a polynomial

f(a) means "replace x by a". If f(x) = x³ − 2x² + 3x − 4, then f(2) = 8 − 8 + 6 − 4 = 2 and f(−1) = −1 − 2 − 3 − 4 = −10. Careful with negative signs: (−1)³ = −1 but (−1)² = +1.

Adding, subtracting and multiplying

Add or subtract by collecting like terms. Multiply by expanding every term of one bracket by every term of the other. The degree of a product is the sum of the degrees.

Worked example 4.1

If f(x) = 2x² − 3x + 1 and g(x) = x − 2, find (a) f(x) + g(x) (b) f(x)g(x).

(a) 2x² − 3x + 1 + x − 2 = 2x² − 2x − 1.
(b) (2x² − 3x + 1)(x − 2) = 2x³ − 4x² − 3x² + 6x + x − 2.
= 2x³ − 7x² + 7x − 2. Degree 3 = 2 + 1, as expected.

2x² − 2x − 1; 2x³ − 7x² + 7x − 2

Worked example 4.2

Given p(x) = 4x³ − x + 5, find p(−2) and p(½).

p(−2) = 4(−8) + 2 + 5 = −32 + 7 = −25.
p(½) = 4(⅛) − ½ + 5 = ½ − ½ + 5 = 5.

−25; 5

⚠️ Common mistake: writing −2² = 4 when substituting x = −2 into x². Use brackets: (−2)² = 4, but −(2²) = −4.

Try it yourself. State the degree and leading coefficient of 5 − 3x² + 7x⁵.

Show solution

Degree 5, leading coefficient 7.

Continue this lesson in the free app →

Practice questions

1. Find the remainder when x³ − 3x² + 4x − 5 is divided by (x − 2).

Show worked answer

8 − 12 + 8 − 5 = −1.

2. Show that (x + 3) is a factor of x³ + 2x² − 5x − 6 and factorise completely.

Show worked answer

f(−3) = −27 + 18 + 15 − 6 = 0. (x − 2)(x + 1)(x + 3).

3. Find k if (x − 3) is a factor of 2x³ − kx² + x + 6.

Show worked answer

54 − 9k + 3 + 6 = 0 → k = 7.

4. Solve x³ − 2x² − 5x + 6 = 0.

Show worked answer

(x − 1)(x + 2)(x − 3) = 0: x = 1, −2, 3.

5. Divide x³ + 4x² − 3 by (x + 1) synthetically.

Show worked answer

1, 3, −3, remainder 0: x² + 3x − 3.

Key words

Polynomial: An expression of the form aₙxⁿ + … + a₁x + a₀ with whole-number powers.
Degree: The highest power of x with a non-zero coefficient.
Coefficient: The number multiplying a power of x.
Leading coefficient: The coefficient of the highest power.
Constant term: The term with no x.
Quotient: The result of a division, Q(x).
Remainder: What is left after division, R.
Root (zero): A value a with f(a) = 0.
Factor: A polynomial that divides f(x) exactly.
Synthetic division: A short method of dividing by (x − a) using coefficients only.
Identity: An equation true for all values of x.
Partial fractions: Splitting a rational expression into simpler fractions.

Learn Remainder and factor theorems step by step, free.
The Cara's Tips app teaches every topic from the very basics, with quick checks, worked examples and exam practice. It works offline.
Open this lesson in the free app Download for Android