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How do you simplify and divide $\left( 6{{\omega }^{5}}-18{{\omega }^{2}}-120 \right)\div \left( \omega -2 \right)$?

seo-qna
Last updated date: 21st Jun 2024
Total views: 375k
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Answer
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Hint: In this question we have to divide the $\left( 6{{\omega }^{5}}-18{{\omega }^{2}}-120 \right)$ by $\left( \omega -2 \right)$. To solve the question we will use the long division method. For this first we will write the all terms in decreasing power of the variable order including the missing terms. Then we start dividing the terms by following the step by step procedure.

Complete step by step answer:
We have been given that $\left( 6{{\omega }^{5}}-18{{\omega }^{2}}-120 \right)\div \left( \omega -2 \right)$.
We have to perform division.
We know that the dividend is divided by divisor to get the quotient.
Here we have dividend $=\left( 6{{\omega }^{5}}-18{{\omega }^{2}}-120 \right)$, divisor $=\left( \omega -2 \right)$.
To solve the given equation we will use the long division method.
For this first we need to arrange the terms in decreasing power of the variable including the missing terms. Missing terms will be written with the coefficient zero so that the meaning of the given expression cannot be changed. Then we will get
$\Rightarrow \left( 6{{\omega }^{5}}+0{{\omega }^{4}}+0{{\omega }^{3}}-18{{\omega }^{2}}+0\omega -120 \right)$
Now, let us start dividing the terms by long division method. Then we will get
\[\omega -2\overset{6{{\omega }^{4}}+12{{\omega }^{3}}+24{{\omega }^{2}}+30\omega +60}{\overline{\left){\begin{align}
  & 6{{\omega }^{5}}+0{{\omega }^{4}}+0{{\omega }^{3}}-18{{\omega }^{2}}+0\omega -120 \\
 & \underline{6{{\omega }^{5}}-12{{\omega }^{4}}} \\
 & 12{{\omega }^{4}} \\
 & 12{{\omega }^{4}}-24{{\omega }^{3}} \\
 & \underline{\overline{\begin{align}
  & 24{{\omega }^{3}}-18{{\omega }^{2}} \\
 & 24{{\omega }^{3}}-48{{\omega }^{2}} \\
\end{align}}} \\
 & 30{{\omega }^{2}} \\
 & \underline{30{{\omega }^{2}}-60\omega } \\
 & 60\omega -120 \\
 & \underline{60\omega -120} \\
 & 0 \\
\end{align}}\right.}}\]
So on dividing the given expression $\left( 6{{\omega }^{5}}-18{{\omega }^{2}}-120 \right)\div \left( \omega -2 \right)$ we get the quotient as \[6{{\omega }^{4}}+12{{\omega }^{3}}+24{{\omega }^{2}}+30\omega +60\].

Note:
In algebraic long division methods we need to follow the same steps as we follow in the arithmetic. We need to perform division until no term is left in the divisor. We can also check our answer by using the formula that $\text{Dividend=quotient}\times \text{divisor+remainder}\text{.}$. So by substituting the values we can verify the answer.