Question

Given expression ${{x}^{4}}+9{{x}^{3}}+35{{x}^{2}}-x+4$ divided by ${{x}^{2}}+10x+29$
 (a) ${{x}^{2}}-x+16$
(b) ${{x}^{2}}-x+16$with a non zero remainder
(c) ${{x}^{2}}-7x+16$
(d) None of these

Answer 1

Hint: In the problem, we are to find the result we get ${{x}^{4}}+9{{x}^{3}}+35{{x}^{2}}-x+4$ divided by ${{x}^{2}}+10x+29$. Starting with this problem, we have to use the long division method to get our result. If we get a remainder as zero, then we get the polynomial is divisible.

Complete step by step answer:
According to the problem, we are given with terms, ${{x}^{4}}+9{{x}^{3}}+35{{x}^{2}}-x+4$ divided by ${{x}^{2}}+10x+29$.
We will try to find what we will get after the division.
To start with, we have the first term as ${{x}^{4}}$ , so multiplying the divisor with ${{x}^{2}}$ will help us to get rid of the first term.
${{x}^{2}}+10x+29\overset{{{x}^{2}}}{\overline{\left){\begin{align}
  & {{x}^{4}}+9{{x}^{3}}+35{{x}^{2}}-x+4 \\
 & {{x}^{4}}+10{{x}^{3}}+29{{x}^{2}} \\
\end{align}}\right.}}$
From this, we get the remainder, $-{{x}^{3}}+6{{x}^{2}}$ and we are adding – x from the term.
Thus, we get, $-{{x}^{3}}+6{{x}^{2}}-x$,
Again, we have the first term as $-{{x}^{3}}$, so multiplying the divisor with – x will help us to get rid of the next term.
${{x}^{2}}+10x+29\overset{-x}{\overline{\left){\begin{align}
  & -{{x}^{3}}+6{{x}^{2}}-x+4 \\
 & -{{x}^{3}}-10{{x}^{2}}-29x \\
\end{align}}\right.}}$
Now, we are getting the remainder as, $16{{x}^{2}}+28x$ and we can add 4 with the term also.
So, we are left with $16{{x}^{2}}+28x+4$.
As we have the first term as, $16{{x}^{2}}$, we will multiply the divisor with 16 to eliminate the first term.
${{x}^{2}}+10x+29\overset{16}{\overline{\left){\begin{align}
  & 16{{x}^{2}}+28x+4 \\
 & 16{{x}^{2}}+160x+435 \\
\end{align}}\right.}}$
This leaves us with a remainder that, 188x – 431.
Hence, we have our dividend as, ${{x}^{2}}-x+16$ and having our remainder as 188x – 431.

So, the correct answer is “Option b”.

Note: We have used the long division method to get the solution in this problem. Here are the steps required for Dividing by a Polynomial Containing More Than One Term (Long Division):
i) Make sure the polynomial is written in descending order. If any terms are missing, use a zero to fill in the missing term (this will help with the spacing). ii) Divide the term with the highest power inside the division symbol by the term with the highest power outside the division symbol. iii) Multiply (or distribute) the answer obtained in the previous step by the polynomial in front of the division symbol. iv) Subtract and bring down the next term. Repeat Steps 2, 3, and 4 until there are no more terms to bring down.