Factor and Remainder Theorem

The Factor and Remainder Theorem form a small, but significant part of the course. It is not at all difficult, but it is worth reminding yourself of these theorem, since, in examinations students often forget them and waste a lot of time using long methods instead. The Factor Theorem is a theorem that allows us to find factors of polynomial functions, to find zeros and ultimately to help us sketch graphs.

A polynomial function f(x) is an algebraic expression that takes the form

\({ f(x)=a }_{ n }{ x }^{ n }+{ a }_{ n-1 }{ x }^{ n-1 }+{ a }_{ n-2 }{ x }^{ n-2 }+\quad ...\quad +{ a }_{ 1 }{ x }^{ 1 }+{ a }_{ 0 }\)

For example, a polynomial function of degree 3 is a cubic function e.g \(f(x) = 7x^3+4x^2-3x+4\)

Key Concepts

On this page, you should learn about

  • polynomial functions
  • zeros, roots and factors
  • the factor and remainder theorem

Essentials

The following videos will help you understand all the concepts from this page

Factorising Polynomials

The Remainder Theorem states that for a polynomial f(x),

the remainder when divided by (x-a) is f(a).

Example

Let \(f(x)=2x^3+4x^2+x-5\)

If we divide this polynomial by \((x-2)\)...

\(\frac{2x^3+4x^2+x-5}{x-2}=2x^2+8x+17+\frac{29}{x-2}\)

... we get a quotient \(2x^2+8x+17\) and a remainder 29

When we work out f(2)...

\(f(2)=2(2)^3+4(2)^2+(2)-5=29\)...

... we get 29


The factor theorem states that for a polynomial f(x),

(x-a) is a factor if and only if f(a)=0

Example

Let \(f(x)=x^4+2x^3-7x^2-8x+12\)

When we work out f(1)...

\(f(1)=(1)^4+2(1)^3-7(1)^2-8(1)+12 = 0\)

If we divide this polynomial by \((x-1)\)...

\(\frac{x^4+2x^3-7x^2-8x+12}{x-1}=x^3+3x^2-4x-12\)

... there is no remainder, \((x-1)\) is a factor of the polynomial


In this video, we will look at an application of the factor theorem which is to factorize polynomials:

Factorize completely \(f(x)=2x^3+x^2-7x-6\)

Notes from the video

Typical Exam-style Question

In the following video, we will look at a typical exam-style question:

The cubic polynomial \(f(x)=ax^3+bx^2-29x+60\) has a factor \((x+4)\) and leaves a remainder of 6 when divided by \((x-2)\).

a. Find the value of a and b.

b. Factorize the polynomial

Notes from the video

Summary

Print from here

Test Yourself

Here is a quiz that practises the skills from this page


START QUIZ!

Exam-style Questions

Question 1

The cubic polynomial \(3x^3+ax^2+bx-12\) has a factor \((x-2)\) and leaves a remainder of -20 when divided by \((x-1)\).

Find the value of a and b.

Hint

Full Solution

Question 2

Given that \(ax^3+bx^2+17x-6\) is exactly divisible by \((x-1)(x-2)\), find the value of a and b.

Hint

Full Solution

Question 3

It is given that \(f(x)=x^3+ax^2+bx+8\)

a. Given that \(x^2-4\) is a factor of \(f(x)\), find the values of a and b.

b. Factorize \(f(x)\) into a product of linear factors.

c. Sketch the graph of \(y=f(x)\) labelling any stationary points and the x and y intercepts.

d. Hence state the range of values of k for which \(f(x)=k\) has exactly one root.

Hint

Full Solution

MY PROGRESS

How much of Factor and Remainder Theorem have you understood?