Question

What must be subtracted from ${{x}^{3}}-6{{x}^{2}}-15x+80$ so that the result is exactly divisible by ${{x}^{2}}+x-12$?

Answer 1

Hint:If we subtract the remainder of ${{x}^{3}}-6{{x}^{2}}-15x+80$ when divided with ${{x}^{2}}+x-12$, then the obtained result will be exactly divisible by ${{x}^{2}}+x-12$. We will use the long division method to divide ${{x}^{3}}-6{{x}^{2}}-15x+80$ with ${{x}^{2}}+x-12$.

Complete step-by-step answer:
It is given in the question that we have to find a polynomial, so that if we subtract it from ${{x}^{3}}-6{{x}^{2}}-15x+80$, the obtained number will exactly be divisible by ${{x}^{2}}+x-12$.
So, from the condition given in the question, we get to know that the dividend should be ${{x}^{3}}-6{{x}^{2}}-15x+80$ and the divisor should be ${{x}^{2}}+x-12$. Now, we will use the long division method to divide ${{x}^{3}}-6{{x}^{2}}-15x+80$ with ${{x}^{2}}+x-12$. It is shown as follows.
${{x}^{2}}+x-12\overset{x-7}{\overline{\left){\begin{align}
  & {{x}^{3}}-6{{x}^{2}}-15x+80 \\
 & \underline{{{x}^{3}}+{{x}^{2}}-12x} \\
 & 0-7{{x}^{2}}-3x+80 \\
 & \underline{0-7{{x}^{2}}-7x+8}4 \\
 & 4x-4 \\
\end{align}}\right.}}$
So, on dividing ${{x}^{3}}-6{{x}^{2}}-15x+80$ with ${{x}^{2}}+x-12$, we get (x - 7) as the quotient and (4x - 4) as the remainder. Now, if we subtract (4x - 4) from ${{x}^{3}}-6{{x}^{2}}-15x+80$, then we will get a polynomial which is exactly divisible by ${{x}^{2}}+x-12$.
Hence, (4x - 4) is the term that must be subtracted from ${{x}^{3}}-6{{x}^{2}}-15x+80$ so that the result is exactly divisible by ${{x}^{2}}+x-12$.

Note: The possible mistake one can make in this question is by considering the quotient obtained after the long division, that is, (x - 7) as the term that has to be subtracted from ${{x}^{3}}-6{{x}^{2}}-15x+80$ so that the result is exactly divisible by ${{x}^{2}}+x-12$. But this is wrong as, the term to be subtracted should be the remainder obtained after the long division that is (4x - 4). Also, the students must be careful while performing the long division as there are chances of calculation mistakes.