Question

What should be added to $8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8$ so that the resulting polynomial is exactly divisible by $4{{x}^{2}}+3x-2$?
(a)$10-14x$
(b)$4x-10$
(c)$3x-5$
(d)$5-3x$

Answer 1

Hint: First of all, we have to divide the given polynomial $8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8$ to $4{{x}^{2}}+3x-2$. After division, you will get some remainder. Now, to get the polynomial which should be added to the given polynomial $8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8$ so that the resulting polynomial is divisible by $4{{x}^{2}}+3x-2$. This means we have to subtract the remainder which we got from the division of two given polynomials from this polynomial $8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8$ to make this polynomial divisible to $4{{x}^{2}}+3x-2$.

Complete step-by-step answer:
We have given a polynomial $8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8$ and we have to add some polynomial to it in such a way that this four degree polynomial becomes divisible by $4{{x}^{2}}+3x-2$.
This clearly shows that, the given polynomial $8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8$ is not exactly divisible by $4{{x}^{2}}+3x-2$ so there must be some remainder when we divide these two polynomials so let us divide the given polynomial $8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8$ by $4{{x}^{2}}+3x-2$.
\[4{{x}^{2}}+3x-2\overset{2{{x}^{2}}+2x-1}{\overline{\left){\begin{align}
  & 8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8 \\
 & 8{{x}^{4}}+6{{x}^{3}}-4{{x}^{2}} \\
 & \dfrac{\begin{matrix}
   - & - & + \\
\end{matrix}}{\begin{align}
  & 0+8{{x}^{3}}+2{{x}^{2}}+7x \\
 & 0+8{{x}^{3}}+6{{x}^{2}}-4x \\
 & \dfrac{\begin{matrix}
   - & - & - & + \\
\end{matrix}}{\begin{align}
  & 0+0-4{{x}^{2}}+11x-8 \\
 & 0+0-4{{x}^{2}}-3x+2 \\
 & \dfrac{\begin{matrix}
   - & - & + & + & - \\
\end{matrix}}{0+0+0+14x-10} \\
\end{align}} \\
\end{align}} \\
\end{align}}\right.}}\]
In the above division of given polynomial $8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8$ by $4{{x}^{2}}+3x-2$ using long division method, we got the remainder as:
$14x-10$
Now, we are going to subtract the above remainder from the given polynomial $8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8$ so that this polynomial will be exactly divisible by $4{{x}^{2}}+3x-2$.
$\begin{align}
  & 8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8-\left( 14x-10 \right) \\
 & 8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8+\left( -14x+10 \right).........Eq.(1) \\
\end{align}$
In the above problem, we are asked to find the polynomial which should be added in the given polynomial $8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8$ so that this polynomial will be exactly divisible by $4{{x}^{2}}+3x-2$.
From eq. (1), we can see that we have added $10-14x$ in the polynomial $8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8$. Hence, the polynomial $10-14x$ is the required polynomial which on addition to the given polynomial $8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8$ and then division to this polynomial $4{{x}^{2}}+3x-2$ will give remainder as 0.

Note: The polynomial $10-14x$ that you got in the above solution when added to the polynomial $8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8$ and then this polynomial will be exactly divisible to $4{{x}^{2}}+3x-2$. We can check whether the added polynomial is actually exactly divisible by $4{{x}^{2}}+3x-2$ or not.
After addition of $10-14x$ to the given polynomial $8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8$ we will get the polynomial as:
$\begin{align}
  & 8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}+7x-8-14x+10 \\
 & =8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}-7x+2 \\
\end{align}$
Now, dividing the above polynomial to $4{{x}^{2}}+3x-2$we get,
\[4{{x}^{2}}+3x-2\overset{2{{x}^{2}}+2x-1}{\overline{\left){\begin{align}
  & 8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}-7x+2 \\
 & 8{{x}^{4}}+6{{x}^{3}}-4{{x}^{2}} \\
 & \dfrac{\begin{matrix}
   - & - & + \\
\end{matrix}}{\begin{align}
  & 0+8{{x}^{3}}+2{{x}^{2}}-7x \\
 & 0+8{{x}^{3}}+6{{x}^{2}}-4x \\
 & \dfrac{\begin{matrix}
   - & - & - & + \\
\end{matrix}}{\begin{align}
  & 0+0-4{{x}^{2}}-3x+2 \\
 & 0+0-4{{x}^{2}}-3x+2 \\
 & \dfrac{\begin{matrix}
   - & - & + & + & - \\
\end{matrix}}{0+0+0+0+0} \\
\end{align}} \\
\end{align}} \\
\end{align}}\right.}}\]
As you can see that, the new polynomial $8{{x}^{4}}+14{{x}^{3}}-2{{x}^{2}}-7x+2$ after addition is exactly divisible to $4{{x}^{2}}+3x-2$.