Question

How do you use synthetic division to divide: $\dfrac{{180x - {x^4}}}{{x - 6}}?$

Answer 1

Hint:Synthetic division is consist of very systematic process, to divide a polynomial expression $a{x^3} + b{x^2} + cx + d$ with $x - q$, you have to go through following steps:
i. Set the synthetic division: equate the denominator or the divisor to get the digit to put in the
division box. And be sure that numerator or dividend is written in decreasing order of the
degree of $x$, if any term is missing then fill $0$ with it as follows:
$
\left. {\underline {\,
q \,}}\! \right| \;a\;b\;c\;d \\
\;\;\;\underline {\;\;\;\;\;\;\;\;\;} \\
$
ii. Now write the first coefficient as it is, as follows
$
\left. {\underline {\,
q \,}}\! \right| \;a\;b\;c\;d \\
\;\;\;\underline {\;\;\;\;\;\;\;\;\;} \\
\;\;\;a \\
$
iii. Multiply the number in the division box with the number which is previously written down
$
\left. {\underline {\,
q \,}}\! \right| \;a\;\;\;b\;\;\;\;c\;\;\;d \\
\;\;\;\underline {\;\;(q \times a)\;\;\;\;\;\;\;} \\
\;\;\;a \\
$
iv. Add $b\;\& \;q \times a$ and write down
v. Again multiply $q$ with the new number which is written and write it and add with next number
vi. Do this until you reach the last number
vii. Write down the final answer, start multiplying variable $x$ with degree less than $1$ from
numerator and decrease one with each next term. The last number is your remainder which will
be written in fraction with original divisor $x - q$ in denominator.

Complete step by step solution:
Given: $\dfrac{{180x - {x^4}}}{{x - 6}}$, rewriting it as
$ = \dfrac{{ - {x^4} + 180x}}{{x - 6}}$
For divisor, $
\Rightarrow x - 6 = 0 \\
\Rightarrow x = 6 \\
$

Setting up the synthetic division
$
= \left. {\underline {\,
6 \,}}\! \right| \; - 1\;\;\;180 \\
\;\;\;\;\;\;\underline {\;\;\;\;\;\;\;\;\;\;\;\;} \\
$
Putting down $ - 1$ as it is
$
= \left. {\underline {\,
6 \,}}\! \right| \; - 1\;\;\;180 \\
\;\;\;\;\;\;\underline {\;\;\;\;\;\;\;\;\;\;\;\;} \\
\;\;\;\;\;\; - 1 \\
$
Multiplying $6$ with $ - 1$ and adding the product with $180$
$
= \left. {\underline {\,
6 \,}}\! \right| \; - 1\;\;\;180 \\
\;\;\;\;\;\;\underline {\;\;\;\;\;\;\; - 6\;} \\
\;\;\;\;\;\; - 1\;\;\;174 \\
$
Now writing the final answer, multiplying $x$ of degree $1$ less in each term and last number having denominator $x - 6$
$
= - 1 \times x + \dfrac{{174}}{{x - 6}} \\
= - x + \dfrac{{174}}{{x - 6}} \\
$
We got our final result $ = - x + \dfrac{{174}}{{x - 6}}$

Note: To use synthetic division for division of polynomials, the denominator must be a linear expression and coefficient of $x$ must be $1$. If it is not $1$ then make it by dividing the expression with coefficient of $x$ Sometimes due to complex expression synthetic division doesn’t work, go for a long division method in that case.