The Factor Theorem (Edexcel IGCSE Maths B): Revision Note

Factor Theorem

What is the factor theorem?

• The factor theorem is used to find the linear factors of a function

  • This is closely related to finding the roots (or solutions) of a function or equation

• For a function $\mathrm{f}\left(x\right)$, the factor theorem tells us that

  • If  $\mathrm{f}\left(a\right)=0$, then $\left(x-a\right)$ is a factor of $\mathrm{f}\left(x\right)$

  •  If $\left(x-a\right)$ is a factor of $\mathrm{f}\left(x\right)$, then  $\mathrm{f}\left(a\right)=0$

How do I use the factor theorem?

• Consider the  function $\mathrm{f}\left(x\right)$ where $\left(x-a\right)$ is a factor

  • Then by the factor theorem we know that $\mathrm{f}\left(a\right)=0$

    • I.e., $x=a$ is a solution to the equation  $\mathrm{f}\left(x\right)=0$

• Or consider the function $\mathrm{f}\left(x\right)$ where $\mathrm{f}\left(a\right)=0$

  • Then by the factor theorem we know  that $\left(x-a\right)$ is a factor of $\mathrm{f}\left(x\right)$

  • Therefore  $\mathrm{f}\left(x\right)=(x-a)\times Q\left(x\right)$

    • where $Q\left(x\right)$ is a function that is also a factor of $\mathrm{f}\left(x\right)$

  • Hence  $\frac{\mathrm{f}\left(x\right)}{\left(x-a\right)}=Q\left(x\right)$

    • I.e. $Q\left(x\right)$ is the quotient when $\mathrm{f}\left(x\right)$ is divided by $\left(x-a\right)$

    • And the remainder is equal to zero

• If the linear factor has a coefficient of x (other than 1) you must first factorise out the coefficient

  • For the linear factor  $(bx – c) =b\left(x-\frac{c}{b}\right)$

    • $\mathrm{f}\left(\frac{c}{b}\right)=0$

    • $\mathrm{f}\left(x\right)=b\left(x-\frac{c}{b}\right)\times Q\left(x\right)=\left(bx-c\right)\times Q\left(x\right)$