Question

Factorize: \[8{{x}^{6}}+95{{x}^{3}}+1\].
A. \[\left( {{x}^{2}}+5x+1 \right)\left( 4{{x}^{4}}-10{{x}^{3}}+23{{x}^{3}}-5x+1 \right)\]
B. \[\left( 2{{x}^{2}}+5x+1 \right)\left( 4{{x}^{4}}-10{{x}^{3}}+23{{x}^{3}}-5x+1 \right)\]
C. \[\left( 2{{x}^{2}}+x+1 \right)\left( 4{{x}^{4}}-10{{x}^{3}}+23{{x}^{3}}-5x+1 \right)\]
D. \[\left( 2{{x}^{2}}+5x+1 \right)\left( 4{{x}^{4}}-10{{x}^{2}}+23{{x}^{3}}-5x+1 \right)\]

Answer 1

Hint: In this problem, we have to factorize the given equation. We can first write the given equation in cubic terms. To solve this problem, we have to know the algebraic cubic formula that is the addition of three cube terms. We can use the addition of cubes formula to expand the given equation and to factorize to get the final answer.

Complete step by step solution:
We know that the given equation to be factored is,
\[8{{x}^{6}}+95{{x}^{3}}+1\]
Now we can assume an equation in cube terms to get the above equation in the formula to be used, we get
\[\Rightarrow {{\left( 2{{x}^{2}} \right)}^{3}}+{{\left( 5x \right)}^{3}}+{{\left( 1 \right)}^{3}}\] ….. (1)
We know that the addition of cubes formula is,
\[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc=\left( a+b+c \right)\left( {{a}^{2}}+{{b}^{2}}+{{c}^{2}}-ab-bc-ca \right)\] .
We can see that from (1),
The value of a = \[2{{x}^{2}}\]
The value of b = 5x
The value of c = 1.
We can substitute the above values in the addition of cubes formula, we get\[\Rightarrow {{\left( 2{{x}^{2}} \right)}^{3}}+{{\left( 5x \right)}^{3}}+1-30{{x}^{3}}=\left( 2{{x}^{2}}+5x+1 \right)\left( {{\left( 2{{x}^{2}} \right)}^{2}}+{{\left( 5x \right)}^{2}}+1-\left( 2{{x}^{2}} \right)5x-5x-2{{x}^{2}} \right)\]
We can now simplify the above step, we get
\[\Rightarrow 8{{x}^{6}}+95{{x}^{3}}+1=\left( 2{{x}^{2}}+5x+1 \right)\left( 4{{x}^{4}}+25{{x}^{2}}+1-10{{x}^{3}}-5x-2{{x}^{2}} \right)\]
Now we can rearrange the above step in descending order of the powers, we get
\[\Rightarrow 8{{x}^{6}}+95{{x}^{3}}+1=\left( 2{{x}^{2}}+5x+1 \right)\left( 4{{x}^{4}}-10{{x}^{3}}+23{{x}^{2}}-5x+1 \right)\]

Therefore, the factors of the equation \[8{{x}^{6}}+95{{x}^{3}}+1\] is an option.
 B. \[\left( 2{{x}^{2}}+5x+1 \right)\left( 4{{x}^{4}}-10{{x}^{3}}+23{{x}^{3}}-5x+1 \right)\].

Note: Students make mistakes while simplifying the step after substituting the value in the addition of cubes formula, which should be done carefully. We should know that, to solve this problem, we have to know the algebraic cubic formula that is the addition of three cube terms. We should remember that the addition of cubes formula is, \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc=\left( a+b+c \right)\left( {{a}^{2}}+{{b}^{2}}+{{c}^{2}}-ab-bc-ca \right)\] .