Question

Find the GCD of the polynomials ${{x}^{4}}+3{{x}^{2}}-x-3$ and ${{x}^{3}}+{{x}^{2}}-5x+3$.

Answer 1

Hint: Here, GCD means greatest common divisor. We have to find the greatest common divisor of ${{x}^{4}}+3{{x}^{2}}-x-3$ and ${{x}^{3}}+{{x}^{2}}-5x+3$. We will first compare the given polynomials. And the smaller polynomial will be selected as the temporary divisor. Using this concept, we will find the greatest common divisor of the given polynomials.

Complete step-by-step answer:

It is given in the question that we have to find the GCD, that is the greatest common divisor of the polynomials ${{x}^{4}}+3{{x}^{2}}-x-3$ and ${{x}^{3}}+{{x}^{2}}-5x+3$. Let us assume $f\left( x \right)={{x}^{4}}+3{{x}^{2}}-x-3$ and $g\left( x \right)={{x}^{3}}+{{x}^{2}}-5x+3$. If we compare f (x) and g (x), we get to know that f (x) > g (x) because the degree of f (x) is greater than that of g (x). So, from f (x) > g (x) we will assume g (x) is the first divisor for f (x). If we divide f (x) with g (x), we get,
\[{{x}^{3}}+{{x}^{2}}-5x+3\overset{x+2}{\overline{\left){\begin{align}
  & {{x}^{4}}+3{{x}^{2}}-x-3 \\
 & \underline{{{x}^{4}}+{{x}^{3}}-5{{x}^{2}}+3x} \\
 & 2{{x}^{3}}+5{{x}^{2}}-4x-3 \\
 & \underline{2{{x}^{3}}+2{{x}^{2}}-10x+6} \\
 & 3{{x}^{2}}+6x-9 \\
\end{align}}\right.}}\]
We get, $3{{x}^{2}}+6x-9$ as the remainder. If we divide it with 3, we get ${{x}^{2}}+2x-3$ which is no equal to 0.
Now, it is clear that ${{x}^{3}}+{{x}^{2}}-5x+3$ is not a divisor of ${{x}^{4}}+3{{x}^{2}}-x-3$. Thus, we will now divide ${{x}^{3}}+{{x}^{2}}-5x+3$ with ${{x}^{2}}+2x-3$. So, we get,
${{x}^{2}}+2x-3\overset{x-1}{\overline{\left){\begin{align}
  & {{x}^{3}}+{{x}^{2}}-5x+3 \\
 & \underline{{{x}^{3}}+2{{x}^{2}}-3x} \\
 & -{{x}^{2}}-2x+3 \\
 & \underline{-{{x}^{2}}-2x+3} \\
 & 0 \\
\end{align}}\right.}}$
We get, (x - 1) as the quotient and 0 as the remainder. It means that ${{x}^{2}}+2x-3$ is a factor of g (x). We also know that ${{x}^{2}}+2x-3$ is a factor of f (x) because we get, ${{x}^{2}}+2x-3$ as the remainder when we divided f (x) with g (x).
It means that ${{x}^{2}}+2x-3$ is the greatest common divisor of f (x) and g (x).
Therefore the GCD of ${{x}^{4}}+3{{x}^{2}}-x-3$ and ${{x}^{3}}+{{x}^{2}}-5x+3$ is ${{x}^{2}}+2x-3$.

Note: The students may make the mistake of dividing the polynomial f (x) with g (x) and they do not get the remainder as 0 and hence may write that there is no such polynomial which divides both the polynomials, but this is wrong as the solution is incomplete. So, the students must do all the calculations and should not miss out anything.