State the Factor Theorem.

If f\left(\frac{b}{a}\right)=0 then (ax-b) is a factor of f(x).

What does f(2)=5 tell you?

The remainder is 5 when f(x) is divided by (x-2).

What do you do after finding one factor of a cubic?

Divide (or compare coefficients) to find the quadratic factor, then factorise it.

Which values should you try first when searching for a root?

\pm1, \pm2, then \frac{b}{a} with b dividing the constant and a dividing the leading coefficient.

How do you find unknown coefficients using a factor and a remainder?

Substitute the root to get f=0 for the factor and f= remainder for the other, then solve simultaneously.

Factor Theorem and Remainder TheoremEdexcel International A Level Maths: Flashcards

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State the Remainder Theorem.

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State the Remainder Theorem.: The remainder when $f(x)$ is divided by $(ax-b)$ is $f\left(\frac{b}{a}\right)$ .
State the Factor Theorem.: If $f\left(\frac{b}{a}\right)=0$ then $(ax-b)$ is a factor of $f(x)$ .
Which value do you substitute to test the factor $(x+3)$ ?: $x=-3$
Which value do you substitute to test the factor $(2x-1)$ ?: $x=\frac12$
What does $f(a)=0$ tell you?: $(x-a)$ is a factor of $f(x)$ .
What does $f(2)=5$ tell you?: The remainder is $5$ when $f(x)$ is divided by $(x-2)$ .
What do you do after finding one factor of a cubic?: Divide (or compare coefficients) to find the quadratic factor, then factorise it.
Which values should you try first when searching for a root?: $\pm1$ , $\pm2$ , then $\frac{b}{a}$ with $b$ dividing the constant and $a$ dividing the leading coefficient.
Factorise $x^3+3x^2-4$ .: $(x-1)(x+2)^2$
Factorise $6x^3+11x^2-x-6$ .: $(x+1)(3x-2)(2x+3)$
How can you show a quadratic factor gives no real roots?: Show that $b^2-4ac<0$ .
How do you find unknown coefficients using a factor and a remainder?: Substitute the root to get $f=0$ for the factor and $f=$ remainder for the other, then solve simultaneously.

Exam questions on Factor Theorem and Remainder Theorem

Let $f(x)=x^3-7x+6$ .
Show that $(x+3)$ is a factor of $f(x)$ .2 marks
Let $g(x)=2x^3+x^2-13x+6$ .
Show that $(x+3)$ is a factor of $g(x)$ .2 marks
Let $f(x)=6x^3+11x^2-3x-2$ .
Show that $(x+2)$ is a factor of $f(x)$ , and find the quadratic $q(x)$ such that $f(x)=(x+2)q(x)$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).