Question

Find the remainder when $ 3{{x}^{3}}-2{{x}^{2}}-x+1 $ is divided by $ x+3 $ .

Answer 1

Hint: Before solving the above question we have to get the idea on the polynomials. The above problem will be solved by using the long division method and remainder theorem. So we are going to use the remainder theorem method to solve the above problem. In the remainder theorem method we are going to find the value of x then substitute it in the given polynomial to find the remainder.

Complete step-by-step answer:
The given polynomial is as follows:
F(x) = $ 3{{x}^{3}}-2{{x}^{2}}-x+1 $ …… (1)
As we can see that the given polynomial is the function of x with the degree of 3 so the polynomial is known as cubic polynomial. So whenever we have to deal with the polynomial with degree more than 3 then we will go with the remainder theorem.
We have to find the value of x the given divisor x + 3 as follows:
 $ \begin{align}
  & x+3=0 \\
 & x=-3
\end{align} $
Now we have to substitute the value of x in expression (1) as follows:
 $ \begin{align}
\Rightarrow & f\left( -3 \right)=3{{\left( -3 \right)}^{3}}-2{{\left( -3 \right)}^{2}}-\left( -3 \right)+1 \\
 & =-95
\end{align} $
Therefore, when the equation $ 3{{x}^{3}}-2{{x}^{2}}-x+1 $ divided by the x + 3 we get the remainder as -95.

Note: We have to be careful in this type of questions because whenever these types of questions come then everyone starts with the long division method to solve these questions and they make the mistakes in doing the simplification of the long division method. So to avoid it we go with the remainder theorem in which we have to be careful only of the positive and negative signs, during the calculation.