Question

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

Answer 1

Hint: We know that the general form of a quadratic equation is $a{{x}^{2}}+bx+c$ since in this question we have a polynomial equation which cannot be factorized directly therefore, in this question we will consider the term ${{x}^{3}}$ as some variable $m$ and then solve the quadratic equation and then simplify the factors further to get the required solution.

Complete step by step solution:
We have the expression as:
$\Rightarrow 72{{x}^{9}}+15{{x}^{6}}+9{{x}^{3}}\to \left( 1 \right)$
We can rearrange the terms in the expression as:
$\Rightarrow 72{{\left( {{x}^{3}} \right)}^{3}}+15{{\left( {{x}^{3}} \right)}^{2}}+9{{x}^{3}}$
Since there is no direct way to factorize the polynomial directly, let’s consider ${{x}^{3}}=m$. On substituting it in the polynomial, we get:
$\Rightarrow 72{{m}^{3}}+15{{m}^{2}}+9m$
On taking the term $m$ common from all the terms, we get:
$\Rightarrow 3m\left( 24{{m}^{2}}+5m+3 \right)$
Now the term $24{{m}^{2}}+5m+3$ is in the form of a quadratic equation therefore we will solve it by splitting the middle term.
As we know, the general form of quadratic equation is $a{{x}^{2}}+bx+c\to (2)$
On comparing equations $(1)$ and $(2)$, we get:
$a=24$
$b=5$
$c=3$
On substituting the values in the formula, we get:
$\Rightarrow ({{m}_{1}},{{m}_{2}})=\dfrac{-5\pm \sqrt{{{5}^{2}}-4(24)(3)}}{2(24)}$
On simplifying the root, we get:
$\Rightarrow ({{m}_{1}},{{m}_{2}})=\dfrac{-5\pm \sqrt{25-288}}{2(24)}$
On simplifying the denominator, we get:
$\Rightarrow ({{m}_{1}},{{m}_{2}})=\dfrac{-5\pm \sqrt{-253}}{48}$
Now since we have the value of determinant less than $0$, it is not possible to express the terms in the form of real factors which implies that the given expression cannot be factorized.

Note: It is to be noted that the expression does have a factorization with quadratic functions but not linear functions. The final factorization would be $3{{x}^{3}}\left( 24{{x}^{6}}+5{{x}^{3}}+3 \right)$. It is to be remembered that a polynomial equation is a combination of variables and their coefficients, the equation given above is a $9th$ degree polynomial equation.
The factors of a polynomial equation are the terms of degree $1$, which have to be multiplied among themselves so that we get the required polynomial equation.