Question

How do you simplify $5{{x}^{6}}\left( 2{{x}^{4}}-{{x}^{3}}+7{{x}^{2}}-4x \right)$?

Answer 1

Hint: To simplify the given expression we will multiply the terms inside the bracket by $5{{x}^{6}}$. Then we combine the like terms obtained by multiplying. Then simplifying the obtained equation we get the desired answer.

Complete step by step solution:
We have been given an expression $5{{x}^{6}}\left( 2{{x}^{4}}-{{x}^{3}}+7{{x}^{2}}-4x \right)$.
We have to simplify the given expression.
Now, to simplify the given expression let us first multiply the terms inside the bracket by $5{{x}^{6}}$. Then we will get
$\Rightarrow 5{{x}^{6}}\times 2{{x}^{4}}-5{{x}^{6}}\times {{x}^{3}}+5{{x}^{6}}\times 7{{x}^{2}}-5{{x}^{6}}\times 4x$
Now, simplifying the above obtained equation we will get
$\Rightarrow 10{{x}^{10}}-5{{x}^{9}}+35{{x}^{8}}-20{{x}^{7}}$
Now, we observe that the coefficients of all x terms in the above obtained equation are divisible by 5. Then by dividing the above obtained equation by 5 we will get
$\Rightarrow \dfrac{10}{5}{{x}^{10}}-\dfrac{5}{5}{{x}^{9}}+\dfrac{35}{5}{{x}^{8}}-\dfrac{20}{5}{{x}^{7}}$
Now, simplifying the above obtained equation we will get
$\Rightarrow 2{{x}^{10}}-{{x}^{9}}+7{{x}^{8}}-4{{x}^{7}}$
As we cannot further simplify the obtained polynomial. Hence above is the required simplified form of the given expression.

Note: As we know that it is difficult to simplify the polynomial having power more than 2. So alternatively we can simplify the given expression by taking the common term out. In the given expression $5{{x}^{6}}\left( 2{{x}^{4}}-{{x}^{3}}+7{{x}^{2}}-4x \right)$ x is the common term inside the bracket. So taking the common term out we will get
$\Rightarrow 5{{x}^{6}}.x\left( 2{{x}^{3}}-{{x}^{2}}+7x-4 \right)$
Now, simplifying the above obtained equation we will get
$\Rightarrow 5{{x}^{7}}\left( 2{{x}^{3}}-{{x}^{2}}+7x-4 \right)$
Hence above is the required simplified form of the given expression.