Question

Perform the division: ${{x}^{4}}-16$ by $x-2$

Answer 1

Hint: To perform the division we will use a long division method. We will simply divide ${{x}^{4}}-16$ by $x-2$ using long division and if we get the remainder as zero that means it is completely divisible by the given value.

Complete step by step answer:
We have to divide the following expression:
${{x}^{4}}-16$…..$\left( 1 \right)$
By the following expression:
$x-2$…..$\left( 2 \right)$
So as we can see equation (1) has first term with power 4 so multiplying the equation (2) by ${{x}^{3}}$ and continuing in the same way till we get the remainder as zero as follows:
$x-2\overset{{{x}^{3}}+2{{x}^{2}}+4x+8}{\overline{\left){\begin{align}
& {{x}^{4}}-16 \\
\end{align}}\right.}}\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\begin{align}
& \underline{-({{x}^{4}}-2{{x}^{3}})} \\
& 0+2{{x}^{3}}-16 \\
& \underline{0-(2{{x}^{3}}-4{{x}^{2}})} \\
& 0+0+4{{x}^{2}}-16 \\
& \underline{0+0-(4{{x}^{2}}-8x)} \\
& 0+0+0+8x-16 \\
& \underline{0+0+0-(8x+16)} \\
& 0+0+0+0+0 \\
\end{align}$
So we get the Quotient as ${{x}^{3}}+2{{x}^{2}}+4x+8$
Hence on performing division ${{x}^{4}}-16$ by $x-2$ we get answer as ${{x}^{3}}+2{{x}^{2}}+4x+8$

Note: Long division is a standard division algorithm which is suitable to divide multiple digit numbers. Long Division is generally used to divide one polynomial by another as it is more accurate and needs calculation which is simple. Other than using long division we can use factoring the expression and cancelling the common term from the two expressions given which is also a simple technique but sometimes it’s not easy to factorize the expression and that may get a little tricky. Some steps that should be taken care of is that both the polynomial should be written in descending order then always divide the term with highest power inside the division with the highest power term outside the division and finally subtract and bring down the next term repeat this process till we get the remainder as zero or till we have brought all the terms down.