Question

How do you use the sum of two cubes formula to factor $x{{y}^{4}}+{{x}^{4}}y?$

Answer 1

Hint: We will take the common terms out and then we will add or subtract some terms, if necessary, so that the given polynomial becomes one with two cubes. It can be then simplified using the identity given by ${{a}^{3}}+{{b}^{3}}=\left( a+b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right).$

Complete step-by-step answer:
Let us consider the given polynomial $x{{y}^{4}}+{{x}^{4}}y.$
We need to factor the given polynomial using the sum of two cubes.
We are going to take the common terms out to make the polynomial in terms of the sum of two cubes.
The common term is $xy.$
Let us take it out to get the polynomial as $xy\left( {{y}^{3}}+{{x}^{3}} \right).$
Let us use the identity for the sum of two cubes here.
We know the identity ${{a}^{3}}+{{b}^{3}}=\left( a+b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right).$
Now, we will apply this identity in the given polynomial in order to find the factors of the polynomial. According to the identity we have written here, the sum of the cubes in the given polynomial will become \[~{{x}^{3}}+{{y}^{3}}=\left( x+y \right)\left( {{x}^{2}}-xy+{{y}^{2}} \right).\]
So, from what we have obtained, we will change the given polynomial as the following polynomial $xy\left( x+y \right)\left( {{x}^{2}}-xy+{{y}^{2}} \right).$
Let us observe the obtained form of the polynomial.
We know that \[x{{y}^{4}}+{{x}^{4}}y=xy\left( x+y \right)\left( {{x}^{2}}-xy+{{y}^{2}} \right).\]
From this, we can understand that $xy$ can divide the given polynomial.
Similarly, we can divide the given polynomial with $x+y.$
We can see that the term \[\left( {{x}^{2}}-xy+{{y}^{2}} \right)\] can also divide the given polynomial.
Hence the terms $xy,x+y$ and \[\left( {{x}^{2}}-xy+{{y}^{2}} \right)\] are the factors of the given polynomial and \[xy\left( x+y \right)\left( {{x}^{2}}-xy+{{y}^{2}} \right)\] is its factored form.

Note: We know that we can use the binomial theorem to simplify the squares, cubes and higher order polynomials. For example, ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ and ${{\left( a+b \right)}^{3}}={{a}^{3}}+3{{a}^{2}}b+3a{{b}^{2}}+{{b}^{3}}.$