Question

How do you factor $3x{{\left( x-2y \right)}^{2}}-{{\left( x-2y \right)}^{3}}$ ?

Answer 1

Hint: The given expression is a polynomial in two variables. First, let us take the common factors out of both terms. The common factor in both terms is a quadratic expression, ${{\left( x-2y \right)}^{2}}$.After taking this term common, evaluate further to check if the expression could be further factorized. Now write all of them together and represent them as the factors of the given expression.

Complete step by step solution:
The given polynomial is $3x{{\left( x-2y \right)}^{2}}-{{\left( x-2y \right)}^{3}}$
It is a polynomial in two variables $x,y\;$
Now let us first take the common terms from both terms.
As we can look through, we see that the term ${{\left( x-2y \right)}^{2}}$ is common in both terms.
It is a quadratic polynomial in two variables.
After taking the common factor out we get,
$\Rightarrow {{\left( x-2y \right)}^{2}}\left[ 3x-\left( x-2y \right) \right]$
Now evaluate the contents in the second term to check if we can still factorize.
$\Rightarrow {{\left( x-2y \right)}^{2}}\left[ 3x-x+2y \right]$
After evaluation we get,
$\Rightarrow {{\left( x-2y \right)}^{2}}\left[ 2x+2y \right]$
Now let us consider the coefficients in front of the variables.
All the coefficients have a common factor $2$ .
Let us take that also out.
After taking the common factor out we get,
$\Rightarrow {{\left( x-2y \right)}^{2}}\times 2\left( x+y \right)$
Rearrange the terms to get,
$\Rightarrow 2{{\left( x-2y \right)}^{2}}\left( x+y \right)$
Now write the squared term in expanded form to get two factors in place of one.
After expanding we get,
$\Rightarrow 2\left( x-2y \right)\left( x-2y \right)\left( x+y \right)$
Hence the factors for the polynomial $3x{{\left( x-2y \right)}^{2}}-{{\left( x-2y \right)}^{3}}$ are $2\left( x-2y \right)\left( x-2y \right)\left( x+y \right)$.

Note: Students tend to make a mistake in thinking $3x$ is also one of the factors in $3x{{\left( x-2y \right)}^{2}}-{{\left( x-2y \right)}^{3}}$ and take it out as a common factor. It is a part of the first term, $3x{{\left( x-2y \right)}^{2}}$ and not common to both the terms. Whenever there is a polynomial that is to be solved, the solution contains the roots of the expression. The number of roots is decided by the degree of the polynomial.