Question

What must be subtracted from $p(x)=8x^{4}+14x^{3}-2x^{2}+7x-8$ so that the resulting polynomial is exactly divisible by $g(x)=4x^{2}+3x-2$ ?

Answer 1

Hint: Try to find out the polynomial nearest to the polynomial $p\left( x \right)$ which is divisible by the polynomial $g\left( x \right)$. Then subtract that polynomial from the polynomial $p\left( x \right)$ to obtain the required solution. All these operations can be done easily by dividing the polynomial $p\left( x \right)$ with the polynomial $g\left( x \right)$. From there the reminder will be the solution.

Complete step by step solution:
Dividing the polynomial $p\left( x \right)=8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8$ with the polynomial $g\left( x \right)=4{{x}^{2}}+3x-2$, we get
$4{{x}^{2}}+3x-2\overset{2{{x}^{2}}+2x-1}{\overline{\left){\begin{align}
  & \text{ }8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8 \\
 & -\left( 8{{x}^{4}}+6{{x}^{3}}-4{{x}^{2}} \right) \\
 & \overline{\begin{align}
  & \text{ }8{{x}^{3}}+2{{x}^{2}}+7x-8 \\
 & \text{ }-\left( 8{{x}^{3}}+6{{x}^{2}}-4x \right) \\
 & \overline{\begin{align}
  & \text{ }-4{{x}^{2}}+11x-8 \\
 & \text{ }-\left( -4{{x}^{2}}-3x+2 \right) \\
 & \overline{\text{ }14x-10} \\
\end{align}} \\
\end{align}} \\
\end{align}}\right.}}$
From the above division, we can conclude that the divisor is $4{{x}^{2}}+3x-2$, the quotient is $2{{x}^{2}}+2x-1$ and the reminder is $14x-10$.
Hence, $14x-10$ should be subtracted from $p\left( x \right)=8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8$ so that the resulting polynomial is exactly divisible by $g\left( x \right)=4{{x}^{2}}+3x-2$.
This is the required solution of the given question.

Note:
From the above division we obtained the divisor as $4{{x}^{2}}+3x-2$ and the quotient as .$2{{x}^{2}}+2x-1$. Multiplying them together, we get
 $\begin{align}
  & \left( 4{{x}^{2}}+3x-2 \right)\left( 2{{x}^{2}}+2x-1 \right) \\
 & \Rightarrow 8{{x}^{4}}+8{{x}^{3}}-4{{x}^{2}}+6{{x}^{3}}+6{{x}^{2}}-3x-4{{x}^{2}}-4x+2 \\
 & \Rightarrow 8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}-7x+2 \\
\end{align}$
Subtracting the above polynomial from the polynomial $p\left( x \right)$, we get
\[\left( 8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8 \right)-\left( 8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}-7x+2 \right)\]
Changing the sign of the terms of the polynomial which is subtracted, we get
\[\begin{align}
  & \Rightarrow 8{{x}^{4}}-8{{x}^{4}}+14{{x}^{3}}-14{{x}^{3}}-2{{x}^{2}}+2{{x}^{2}}+7x+7x-8-2 \\
 & \Rightarrow 14x-10 \\
\end{align}\]
Here also we are getting the same answer.
Hence, $14x-10$ should be subtracted from $p\left( x \right)=8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8$ so that the resulting polynomial is exactly divisible by $g\left( x \right)=4{{x}^{2}}+3x-2$.
This is the verification that the division was correct.