Factor & Remainder Theorems (Edexcel IGCSE Further Pure Maths): Revision Note

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  $\mathrm{f}\left(a\right)=0$

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

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

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

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

The remainder theorem is used to find the remainder when we divide a polynomial function by a linear function

When a polynomial function $\mathrm{f}\left(x\right)$ is divided by a linear function $\left(x-a\right)$, the value of the remainder $R$ is given by $\mathrm{f}\left(a\right)=R$

• Note, if $\mathrm{f}\left(a\right)=0$ then $\left(x-a\right)$ is a factor of $\mathrm{f}\left(x\right)$; this is the factor theorem

Consider a polynomial function $\mathrm{f}\left(x\right)$ and a linear function  $\left(x-a\right)$

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

  • $Q\left(x\right)$ is the quotient (also a polynomial function)

  • $R$ is the remainder (a real number)

• This may also be written as $\mathrm{f}\left(x\right)=Q\left(x\right)\times \left(x\mathit{-}a\right)+R$

• The remainder theorem tells us that  $R=\mathrm{f}\left(a\right)$

  • I.e. we don't need to do the algebraic division to find the remainder!

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

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

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