Question

How do you long divide\[\dfrac{5{{x}^{4}}-2{{x}^{3}}-7{{x}^{2}}-39}{{{x}^{2}}+2x-4}\]?

Answer 1

Hint: In order to long divide polynomials of degree 4 with another polynomial of degree 2 we have to first arrange the polynomial in the decreasing order, use zero in the place of missing terms. Divide highest degree term of dividend by the highest degree term of divisor and multiply this result with the divisor and subtract this with the dividend. Repeat this process until remainder becomes zero or degree of remainder is smaller than divisor .

Complete step-by-step answer:
We are given a polynomial of degree 4 which we have to divide with another polynomial of degree 2 .
As we can see our dividend is equal to \[5{{x}^{4}}-2{{x}^{3}}-7{{x}^{2}}-39\]and our divisor is equal to \[{{x}^{2}}+2x-4\].
When we divide a quartic polynomial with a quadratic polynomial, we get a quadratic quotient.
For the long division method, there are some steps which should be followed while dividing the polynomial.
Step 1: Very first make sure that the polynomial which we are going to divide is written in descending order in other words terms having higher degree written first .If in case ,there is any term missing ,use zero to fill the missing place.
Step 2 : Now divide the term having the highest degree inside the division symbol(dividend) with the highest degree term of the polynomial written outside the divide symbol(divisor) and append it in the quotient.
Step 3: Use the result obtained in step 2 to multiply it with the divisor.
Step 4: Subtract the result in Step 3 with the polynomial inside the divide symbol and bring the remaining terms.
Step 5: Repeat Steps no 2,3 and 4 until no more terms are left to bring down.
Step 6 : Quotient is the final answer.

\[{{x}^{2}}+2x-4\overset{5{{x}^{2}}-12x+37}{\overline{\left){\begin{align}
  & 5{{x}^{4}}-2{{x}^{3}}-7{{x}^{2}}+0.x-39 \\
 & \underline{5{{x}^{4}}+10{{x}^{3}}-20{{x}^{2}}} \\
 & -12{{x}^{3}}+13{{x}^{2}}+0.x-39 \\
 & \underline{-12{{x}^{3}}-24{{x}^{2}}+48x} \\
 & 37{{x}^{2}}-48x-39 \\
 & \underline{37{{x}^{2}}+74x-148} \\
 & -122x+109 \\
\end{align}}\right.}}\]
We first tried to equate the highest power of the dividend with the highest power of the divisor and that’s why we multiplied with $ 5{{x}^{2}} $ . We get $ 5{{x}^{4}}+10{{x}^{3}}-20{{x}^{2}} $ . We subtract it to get $ 12{{x}^{3}}+13{{x}^{2}}+0.x-39 $ . We again equate with the highest power of the remaining terms. We multiply with $ -12x $ and subtract to get \[37{{x}^{2}}-48x-39\]. At the end we had to multiply with 37 to get \[-122x+109\] as remainder. The quotient is $ 5{{x}^{2}}-12x+37 $ .
Therefore, long division of\[\dfrac{5{{x}^{4}}-2{{x}^{3}}-7{{x}^{2}}-39}{{{x}^{2}}+2x-4}\]is equal to $ 5{{x}^{2}}-12x+37 $ with remainder $ -122x+109 $ .

Note: 1.Stop the process of long division when the highest degree of divisor is greater than that of remainder.
2.The polynomial who is getting divided is the dividend and the one who is going to divide is the divisor.