Question

How do you find the factors of \[{{x}^{3}}-81\]?

Answer 1

Hint: This type of question is based on the concept of factorization. We can solve this question with the help of factorization of cubic terms, that is,\[{{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)\]. This formula is normally used to find the factors of cubic terms in a single equation. Now, convert 81 into a cubic term, that is, \[81={{9}^{2}}={{\left( {{9}^{\dfrac{2}{3}}} \right)}^{3}}\]. Here, a=x and b=\[{{9}^{\dfrac{2}{3}}}\]. We can find the factors of \[{{x}^{3}}-81\]using the formula \[{{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)\] .

Complete step by step answer:
According to the question, we have been given the expression as \[{{x}^{3}}-81\].
We need to find the factors of \[{{x}^{3}}-81\].
First, we have to find the cubic term of 81.
We know that \[81={{9}^{2}}\].
Therefore, by multiplying 3 in the numerator and the denominator, we get,
\[81={{9}^{\dfrac{2}{3}\times 3}}\]
which is as same as,
\[81={{\left( {{9}^{\dfrac{2}{3}}} \right)}^{3}}\]
Now we get,
\[{{x}^{3}}-81={{x}^{3}}-{{\left( {{9}^{\dfrac{2}{3}}} \right)}^{3}}\]
Now, let us consider a=x and b=\[{{9}^{\dfrac{2}{3}}}\].
We will put the values in the formula \[{{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)\].
Therefore, the factors of \[{{x}^{3}}-81\] are
\[{{x}^{3}}-81=\left( x-{{9}^{\dfrac{2}{3}}} \right)\left( {{x}^{2}}+x{{9}^{\dfrac{2}{3}}}+{{9}^{\dfrac{2}{3}\times 2}} \right)\].
On further simplification, we get,
\[{{x}^{3}}-81=\left( x-{{9}^{\dfrac{2}{3}}} \right)\left( {{x}^{2}}+{{9}^{\dfrac{2}{3}}}x+{{9}^{\dfrac{4}{3}}} \right)\]
Therefore, the factors are \[\left( x-{{9}^{\dfrac{2}{3}}} \right)\] and \[\left( {{x}^{2}}+{{9}^{\dfrac{2}{3}}}x+{{9}^{\dfrac{4}{3}}} \right)\].
Hence, the factors of \[{{x}^{3}}-81\] are \[\left( x-{{9}^{\dfrac{2}{3}}} \right)\] and \[\left( {{x}^{2}}+{{9}^{\dfrac{2}{3}}}x+{{9}^{\dfrac{4}{3}}} \right)\].

Note:
Whenever we get this type of problem, we need to make sure about the formula used for factorization. In this question, we use the factorization for cubic terms. Also, we should avoid calculation mistakes to obtain accurate answers. Similarly, the factors of square terms can also be obtained.
The final answer of the solution can be obtained in another way. Substitute \[{{9}^{2}}={{3}^{4}}\] everywhere in the above solution. Then equate to the final solution. Therefore, the factors of the given equation \[{{x}^{3}}-81\] is \[\left( x-{{3}^{\dfrac{4}{3}}} \right)\] and \[\left( {{x}^{2}}+{{3}^{\dfrac{4}{3}}}x+{{3}^{\dfrac{8}{3}}} \right)\].