Question

How do you factor $3{{x}^{3}}+2{{x}^{2}}-27x-18$?

Answer 1

Now to factor the given expression we will first simplify the expression by taking ${{x}^{2}}$ common from the first two terms and $9$ common from the last two terms. Now we will further simplify the expression and use the formula ${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$ to find all the factors of the given expression.

Complete step by step solution:
Now consider the given expression $3{{x}^{3}}+2{{x}^{2}}-27x-18$
The above expression is a cubic equation in x.
Now we want to factor the above expression.
To factor the given expression we will first simplify the given expression.
Now to simplify the expression we will group common terms in the expression.
Hence we will take ${{x}^{2}}$ common from the first two terms and 9 common from the last two terms. Hence we get,
$\Rightarrow {{x}^{2}}\left( 3x+2 \right)-9\left( 3x+2 \right)$
Now taking $3x+2$ from the whole expression we get,
$\Rightarrow \left( 3x+2 \right)\left( {{x}^{2}}-9 \right)$
Now we have a quadratic expression ${{x}^{2}}-9$ . Hence now we will factorize the quadratic.
Let us first write ${{x}^{2}}-9={{x}^{2}}-{{3}^{2}}$ .
Now we know that ${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$
Hence using this we get ${{x}^{2}}-{{3}^{2}}=\left( x-3 \right)\left( x+3 \right)$ .
Hence again substituting this in the expression we get,
$\Rightarrow \left( 3x+2 \right)\left( x-3 \right)\left( x+3 \right)$
Hence the factors of the given expression are $\left( 3x+2 \right)$ , $\left( x-3 \right)$ and $\left( x+3 \right)$

Note: Now note that to find the factors of the expression we can also try to find the roots of the expression. Now to find the roots of the cubic equation we will try to substitute different values of x and hence find the first root. Hence we will write the factor corresponding to the root of the expression. Now divide the whole expression by the factor to form a quadratic. Again find the roots of the quadratic and hence factor the expression.