Question

How do you divide $\dfrac{8{{x}^{3}}+5{{x}^{2}}-12x+10}{{{x}^{2}}-3}$?

Answer 1

Hint: We are given a polynomial $8{{x}^{3}}+5{{x}^{2}}-12x+10$ we are asked to divide it by ${{x}^{2}}-3$. To do so, we will start our solution by identifying the numerator and the denominator and see that the polynomials in the numerator and denominator are arranged in decreasing order of their power for long division. To solve this question, we will learn about the long division method and will apply the steps to perform the long division of the given polynomials.

Complete step by step answer:
We are given, $\dfrac{8{{x}^{3}}+5{{x}^{2}}-12x+10}{{{x}^{2}}-3}$. It’s a fraction consisting of two polynomials. $8{{x}^{3}}+5{{x}^{2}}-12x+10$, a 3 degree polynomial in the numerator and ${{x}^{2}}-3$, a 2 degree polynomial in the denominator. To divide the polynomial, we should check that the polynomial in the numerator and denominator are arranged in decreasing order of their power.
For example, consider $\dfrac{4+{{x}^{2}}+2{{x}^{3}}}{{{x}^{2}}+x}$. Here, the numerator is not well defined, so we will arrange it in a descending order based on their power. So, we will get the numerator after arranging it as $2{{x}^{3}}+{{x}^{2}}+4$. Now, we will consider our question, so we have $\dfrac{8{{x}^{3}}+5{{x}^{2}}-12x+10}{{{x}^{2}}-3}$. In our polynomial both the numerator and the denominator are in the correct order. Now, we will learn how to perform the long division of polynomials.
Step 1: Divide the first term of the numerator by the first term of the denominator and then put that in the quotient.
Step 2: Multiply the denominator by that value and then put that below the numerator.
Step 3: Now, subtract them to create the next polynomial.
Step 4: Repeat these steps again with the new polynomial.
Now, we have $\dfrac{8{{x}^{3}}+5{{x}^{2}}-12x+10}{{{x}^{2}}-3}$. The first term of the numerator is $8{{x}^{3}}$ and that of the denominator is ${{x}^{2}}$. So, on dividing them, we get $\dfrac{8{{x}^{3}}}{{{x}^{2}}}=8x$.
Now, we will multiply the denominator by $8x$ and put it below the numerator.
\[{{x}^{2}}-3\overline{\left){\begin{align}
  & 8{{x}^{3}}+5{{x}^{2}}-12x+10 \\
 & 8{{x}^{3}}-24x \\
 & \overline{5{{x}^{2}}+12x+10} \\
\end{align}}\right.}\left( 8x \right.\]
Now, the largest degree term in numerator is $5{{x}^{2}}$ and in the denominator we have, ${{x}^{2}}$. So, we will now divide $5{{x}^{2}}$ by ${{x}^{2}}$, so we get,
$\dfrac{5{{x}^{2}}}{{{x}^{2}}}=5$ So, we will multiply the denominator by 5 and put it below the numerator, so we get,
\[{{x}^{2}}-3\overline{\left){\begin{align}
  & 8{{x}^{3}}+5{{x}^{2}}-12x+10 \\
 & 8{{x}^{3}}-24x \\
 & \overline{\begin{align}
  & 5{{x}^{2}}+12x+10 \\
 & 5{{x}^{2}}-15 \\
 & \overline{12x+25} \\
\end{align}} \\
\end{align}}\right.}\left( 8x+5 \right.\]
Now, we are left with a term in the numerator with degree less than that of the degree of denominator, so it cannot be divided further. Hence, our solution is 8x+5 and the remainder is 12x+25.

Note:
Students have to be careful while subtracting two values, as the sign of the terms will get changed, so if we use the wrong sign at some point in the division, then our solution will become wrong. We can divide higher power by lower power terms and if we are left with the denominator with higher degree than the numerator, then division will stop.