4.6 The factor and remainder theorems
The remainder theorem states that for any number , the remainder when we divide a polynomial (which can be written as ) by is equal to . 66 6 We can prove this without too much trouble. Firstly, by the definition of division, we can write the result of any division as a ”quotient” and a remainder as a fraction of - as an equation, that is, that . If we multiply both sides by , then we obtain that . We can then set , which gives that . Note that .
A special case of the remainder theorem is known as the "factor theorem" and it relates the roots of a polynomial to its factors. If divides with no remainder, then by the remainder theorem, this means that and so is a root as well as a factor. Roots are factors, and factors are roots.
4.6.1 An example
: Given that is a factor of , find the value of .
: We can directly apply the factor theorem here; as is a factor of , it must be the case that . With this information, we can form an equation where the only variable is .
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