Question

How do you find the product of $\left( x-y \right)\left( {{x}^{2}}+xy+{{y}^{2}} \right)$ ?

Answer 1

Hint: In this question, we have to find the product of an algebraic expression. Thus, we will use the distributive property to get the solution. First, we will apply the distributive property $\left( a-b \right)c=ac-bc$ in given expression. After that, we will again apply the distributive property $a\left( b+c+d \right)=ab+ac+ad$ in the expression to get a more simplified term. In the last, we will make the necessary calculations by opening the brackets of the algebraic expression and then cancel out the same terms with opposite signs, to get the required solution for the problem.

Complete step by step answer:
According to the problem, we have to find the product of the given algebraic expression.
Thus, we will use the distributive property to get the solution.
The algebraic expression given to us is $\left( x-y \right)\left( {{x}^{2}}+xy+{{y}^{2}} \right)$ -------- (1)
First, we will apply the distributive property $\left( a-b \right)c=ac-bc$ in the algebraic expression (1), we get
$\Rightarrow x\left( {{x}^{2}}+xy+{{y}^{2}} \right)-y\left( {{x}^{2}}+xy+{{y}^{2}} \right)$
Now, we will again apply the distributive property $a\left( b+c+d \right)=ab+ac+ad$ in the above expression, we get
$\Rightarrow x\left( {{x}^{2}} \right)+x\left( xy \right)+x\left( {{y}^{2}} \right)-y\left( {{x}^{2}} \right)-y\left( xy \right)-y\left( {{y}^{2}} \right)$
Now, we will open the brackets of the above algebraic expression, we get
$\Rightarrow {{x}^{3}}+{{x}^{2}}y+x{{y}^{2}}-{{x}^{2}}y-x{{y}^{2}}-{{y}^{3}}$
As we know, the same terms with opposite signs cancel out each other, therefore we get
$\Rightarrow {{x}^{3}}-{{y}^{3}}$ which is the required answer.

Therefore, the product of $\left( x-y \right)\left( {{x}^{2}}+xy+{{y}^{2}} \right)$ is equal to ${{x}^{3}}-{{y}^{3}}$.

Note: While solving this problem, do mention all the steps properly to avoid confusion and mathematical error. One of the alternative methods to solve this problem is using the algebraic identity ${{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)$ to get an accurate solution for the problem. Always mention all the properties carefully to avoid an error.