Question

How do you simplify ${{(5-2{{x}^{3}})}^{2}}$?

Answer 1

Hint: We will solve the given expression by splitting the given expression into its exclusive terms, which is, ${{(5-2{{x}^{3}})}^{2}}=(5-2{{x}^{3}})(5-2{{x}^{3}})$ and now after having two sets of parentheses, we will now multiply each component present in one set of parenthesis to the each component of the other, further solving the resulting expression will give the required result.

Complete step by step solution:
According to the question given, we have an expression having power raised to 2. We will first factorize it into its component terms and then continue with the multiplication.
So we have,
${{(5-2{{x}^{3}})}^{2}}$
We can write this as,
${{(5-2{{x}^{3}})}^{2}}=(5-2{{x}^{3}})(5-2{{x}^{3}})$
We will now multiply each component present in one set of parentheses to each component of the other set of parenthesis, we get,
\[\Rightarrow 5(5-2{{x}^{3}})-2{{x}^{3}}(5-2{{x}^{3}})\]
5 will be multiplied to each component present in \[(5-2{{x}^{3}})\] and in the same way \[-2{{x}^{3}}\] will be multiplied to each component present in \[(5-2{{x}^{3}})\].
On multiplication, we get,
\[\Rightarrow (25-10{{x}^{3}})-(10{{x}^{3}}-4{{x}^{6}})\]
On opening the brackets, similar terms will get cancelled and we get,
\[\Rightarrow 25-10{{x}^{3}}-10{{x}^{3}}+4{{x}^{6}}\]
Adding up the similar terms we get,
\[\Rightarrow 25-20{{x}^{3}}+4{{x}^{6}}\]
Rearranging we get,
\[\Rightarrow 4{{x}^{6}}-20{{x}^{3}}+25\]
So, this is the required result of the multiplication.

Note: The given expression when written in component form the components inside the parenthesis should not change else might cause errors. While multiplying each term, multiplication operation should be done carefully so as not to write any wrong answers. After solving the expression, the final answer should be written in the decreasing order of the degree of the polynomial.