Question

The polynomial \[p\left( x \right)=a{{x}^{3}}+4{{x}^{2}}+3x-4\] and \[q\left( x \right)={{x}^{3}}-4x+a\] leave same remainder when divided by \[\left( x-3 \right)\]. Find a and hence find the remainder when \[p\left( x \right)\] is divided by \[\left( x-2 \right)\].

Answer 1

Hint: For solving this question you should know about finding the remainder by remainder theorem directly. In this problem we can also find the remainder by the help of a long division method but that will be a long method. So, we use the remainder theorem here for solving this.

Complete step by step answer:
According to our question it is asked to us that the polynomial \[p\left( x \right)=a{{x}^{3}}+4{{x}^{2}}+3x-4\] and \[q\left( x \right)={{x}^{3}}-4x+a\] leave same remainder when divided by \[\left( x-3 \right)\]. Find a and hence find the remainder when \[p\left( x \right)\] is divided by \[\left( x-2 \right)\].
So, as we know that we can calculate the remainder by the long division method in this problem and that is a big method for this, so we will use the remainder theorem here for solving this problem.
Given, \[a{{x}^{3}}+4{{x}^{2}}+3x-4=0\] & \[{{x}^{3}}-4x+a=0\] leave same remainder when divided by \[x-3\].
\[p\left( x \right)=a{{x}^{3}}+4{{x}^{2}}+3x-4\]
\[q\left( x \right)={{x}^{3}}-4x+a\]
Remainder theorem,
\[\begin{align}
  & p\left( 3 \right)=q\left( 3 \right) \\
 & a{{\left( 3 \right)}^{3}}+4{{\left( 3 \right)}^{2}}+3\left( 3 \right)-4={{3}^{3}}-4\left( 3 \right)+a \\
 & 27a+36+9-4=27-12+a \\
 & 26a=15-41 \\
 & 26a=-26 \\
 & \therefore a=-1 \\
\end{align}\]
And if substitute values here,
\[p\left( x \right)=-{{x}^{3}}+4{{x}^{2}}+3a-4\]
When divided by \[\left( x-2 \right)\]
\[p\left( 2 \right)=-{{\left( 2 \right)}^{3}}+4{{\left( 2 \right)}^{2}}+3\left( 2 \right)-4\]
           \[\begin{align}
  & =-8+16+6-4 \\
 & =8+2 \\
\end{align}\]
            \[=10\]
So, the remainder is 10.

Note: While solving this type of questions you should be aware that if we want to get quotients also with the remainder then we will calculate the remainder by long division method for that. And if we don’t want to calculate quotients then use the remainder theorem for getting the remainder.