Question

How do you factor $3{{x}^{4}}-12{{x}^{3}}$?

Answer 1

Hint: Now we are given with a polynomial in degree 4 in x. To find the roots of the expression we will first take ${{x}^{3}}$ common from the whole expression. Now we will further simplify the expression and hence we will get the expression in terms of factors. Hence the given expression is factored.

Complete step by step solution:
Now we are given with a polynomial in x of degree 4. We want to factor the whole polynomial.
Now factors of polynomials are linear expression which can divide the whole polynomial without leaving any remainder. Hence we want to find all such linear factors of the given polynomial.
Now we can see that there is no constant in the expression.
Hence we can take x common from the whole expression. Now here in the given expression the highest power of x common in all terms is ${{x}^{3}}$
Hence we will take ${{x}^{3}}$ common from the expression.
$\Rightarrow {{x}^{3}}\left( 3x-12 \right)$
Hence the given equation is factored and the factors of the given expression is $x,x,x$ and $3x-12$ .
Also note that the expression is nothing but a product of all the factors.
Hence we get ${{x}^{3}}\left( 3x-12 \right)=3{{x}^{4}}-12{{x}^{3}}$

Note: Now note that we can easily find the roots by factoring the expression. Similarly if we have the roots of the expression we can write the factors of the expression. Now if $\alpha $ and $\beta $ are the roots of the expression then $x-\alpha $ and $x-\beta $ are the factors of the expression. Similarly if $x-\alpha $ and $x-\beta $ then on substituting the value of x as $\alpha $ or $\beta $ we get zero.
Hence we get the roots of the expression.