Question

How do you factor $72{x^9} + 15{x^6} + 9{x^3}$?

Answer 1

As the given equation is quadratic in one variable, we will use the factorization method to find the factors of the given equation. First, take common words from each term. Now, factorize the given equation using splitting the middle term. Check for the two factors of 72 such that when they are added or subtracted, you will get 5. If the above conditions are met, take common and make factors. If not, the equation is in factored form.

Complete step-by-step answer:
We have been given an equation $72{x^9} + 15{x^6} + 9{x^3}$.
We have to find the factors of the given equation.
Take out the common from each term,
$\Rightarrow 72{x^9} + 15{x^6} + 9{x^3} = 3{x^3}\left( {24{x^6} + 5{x^3} + 3} \right)$
The next step is to find two factors of 72 in such a way that when those factors are added or subtracted, we get 5. There are no two such factors.

Hence, the factors of the equation $72{x^9} + 15{x^6} + 9{x^3}$ is $3{x^3}\left( {24{x^6} + 5{x^3} + 3} \right)$.

Note:
As we know the form of quadratic equation in two variables is ${x^2} - \left( {\alpha + \beta } \right)xy + {y^2} = 0$ or ${x^2} -$(Sum of roots)$xy +$Product of roots $= 0$. The factorization method uses the same concept. For the verification of splitting, we can check the brackets formed of factorization. If the two brackets formed after taking common parts from the first two terms and last two terms; these should be equal. If the two brackets formed are not equal, then splitting has gone wrong and we need to check the splitting step once again. Students make mistakes during taking common between the first two and last two terms. We need to take the common HCF of the first two and last two terms respectively, then we will get two brackets formed which is equal.