Question

How do I use the remainder theorem to evaluate polynomials?

Answer 1

Hint: Here we are asked to find out the method to evaluate the polynomials by using the remainder theorem. Firstly, we learnt about the basic definition of the remainder theorem and the formula to be used. And then, we are going to know how to evaluate polynomials using the remainder theorem with an example.

Complete step by step answer:
Remainder theorem is defined as when a polynomial f(x) is divided by the factor (x-a) when the factor is not necessarily an element of the polynomial, then you find a smaller polynomial along with a remainder. The resultant obtained is the value of the polynomial f(x) where x=a and this is possible only if f(a)=0. In order to factorize easily, the remainder theorem is applied.
For example: If a polynomial \[f(x)={{x}^{3}}+3{{x}^{2}}+3x+1\] is divided by \[x+1\]., the remainder is to be find out.
We have to evaluate the given equation at \[x=-1\]. Since, we have to proceed using the remainder theorem,
Here, we can see that polynomial \[f(x)={{x}^{3}}+3{{x}^{2}}+3x+1\] is getting divided so that it will be the dividend.
Also, we can see that the polynomial \[x+1\] is used to divide so it is the divisor.
Now, let us do the long division process then we get,
\[x+1\overline{\left){{{x}^{3}}+3{{x}^{2}}+3x+1}\right.}\]
Now, we know that the quotient is obtained by dividing the first term of dividend with the first term of divisor, similarly second term with second term of divisor, and so on.
\[x+1\overset{{{x}^{2}}+2x+1}{\overline{\left){\begin{align}
  {{x}^{3}}+3{{x}^{2}}+3x+1 & \\
  \underline{{}_{-}\left( {{x}^{3}}+{{x}^{2}} \right)\text{ }} & \\
  2{{x}^{2}}+3x+1 & \\
  \underline{{}_{-}\left( 2{{x}^{2}}+2x \right)\text{ }} & \\
  x+1 & \\
  \underline{{}_{-}\left( x+1 \right)} & \\
  0 & \\
\end{align}}\right.}}\]Here, we can see that the degree of remainder is 0 which is less than that of the divisor which says that the division is complete.
Hence, we can conclude that the remainder of the polynomial is ‘0’.
By the remainder theorem we can say that if x+1 is the factor of f(X) then we can write f(-1)=0 which proves the remainder theorem.

Note: Remainder theorem is a simple and easy way to find the remainder when a polynomial function is divided by another polynomial function. Such questions require simple changes of variables and can be solved easily by keeping in mind the algebraic rules such as substitution and transportation. The long division should be done without any mistakes.