Factor & Remainder Theorem (Cambridge (CIE) O Level Additional Maths): Revision Note

Factor theorem

What is the factor theorem?

• The factor theorem is a useful result concerning the roots and factors of polynomials

  • In the example below, the polynomial $4x^3+8x^2-9x-18$ has three (linear) factors

    • $\left(x+2\right),\left(2x+3\right)$ and $\left(2x-3\right)$

    • and so it has the three roots $x=-2,x=-\frac{3}{2}$ and $x=\frac{3}{2}$

• For a polynomial $\mathrm{f}\left(x\right)$ the factor theorem states that:

i) if $\mathrm{f}\left(p\right)=0$, then $\left(x-p\right)$ is a factor of $\mathrm{f}\left(x\right)$
($x=p$ is a root of $\mathrm{f}\left(x\right)$)

and

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

Remainder theorem

What is the remainder theorem?

• The factor theorem is actually a special case of the more general remainder theorem

• The remainder theorem states that when the polynomial $\mathrm{f}\left(x\right)$ is divided by $\left(x-a\right)$ the remainder is $\mathrm{f}\left(a\right)$

  • You may see this written formally as $\mathrm{f}\left(x\right)=\left(x-a\right)\mathrm{Q}\left(x\right)+\mathrm{f}\left(a\right)$

  • In polynomial division

    • $\mathrm{Q}\left(x\right)$ would be the result (at the top) of the division (the quotient)

    • $\mathrm{f}\left(a\right)$ would be the remainder (at the bottom)

    • $\left(x-a\right)$ is called the divisor

  • In the case when $\mathrm{f}\left(a\right)=0,\mathrm{f}\left(x\right)=\left(x-a\right)\mathrm{Q}\left(x\right)$ and hence $\left(x-a\right)$ is a factor of $\mathrm{f}\left(x\right)$ – the factor theorem!

How do I solve problems involving the remainder theorem?

• If it is the remainder that is of particular interest, the remainder theorem saves the need to carry out polynomial division in full

  • e.g.  The remainder from ${(x}^2-2x)÷(x-3)$ is $3^2-2\times 3=3$

  • This is because if $\mathrm{f}\left(x\right)=x^2-2x$ and $a=3$

• If the remainder from a polynomial division is known, the remainder theorem can be used to find unknown coefficients in polynomials

  • g. The remainder from $(x^2+px)÷(x-2)$ is 8 so the value of p can be found by solving $2^2+p\left(2\right)=8$, leading to $p=2$

  • In harder problems there may be more than one unknown in which case simultaneous equations would need setting up and solving

• The more general version of remainder theorem is if $\mathrm{f}\left(x\right)$ is divided by $\left(ax-b\right)$ then the remainder is  $\mathrm{f}\left(\frac{b}{a}\right)$

  • The remainder is still found by evaluating the polynomial at the value of $x$ such that $ax-b=0$ (the divisor is zero) but it is not necessarily an integer