Question

Factorise: $x\left( {{y}^{2}}-{{z}^{2}} \right)+y\left( {{z}^{2}}-{{x}^{2}} \right)+z({{x}^{2}}-{{y}^{2}})$ .

Answer 1

Hint: In Mathematics, factorization or factoring is defined as the breaking or decomposition of an polynomial into a product of another polynomial which is simpler and of lesser degree than the previous one. In this problem, first we expand the given term and then try to rearrange the term to get common terms from the whole expression. This approach will help in solving the problem.

Complete step-by-step answer:
In the factorization method, we reduce any algebraic or quadratic equation into its simpler form, where the equations are represented as the product of factors instead of expanding the brackets. The factors of any equation can be an integer, a variable or an algebraic expression itself. We use this concept in solving our question.
According to the problem statement, we are given:
$\Rightarrow x\left( {{y}^{2}}-{{z}^{2}} \right)+y\left( {{z}^{2}}-{{x}^{2}} \right)+z({{x}^{2}}-{{y}^{2}})$
On solving the brackets of the expression, we get:
$\Rightarrow x{{y}^{2}}-x{{z}^{2}}+y{{z}^{2}}-y{{x}^{2}}+z{{x}^{2}}-z{{y}^{2}}$
Now, arrange the terms in descending order of power of x.
$\begin{align}
  & \Rightarrow -{{x}^{2}}y+z{{x}^{2}}+x{{y}^{2}}-x{{z}^{2}}-z{{y}^{2}}+y{{z}^{2}} \\
 & \Rightarrow -{{x}^{2}}\left( y-z \right)+x\left( {{y}^{2}}-{{z}^{2}} \right)-yz\left( y-z \right) \\
\end{align}$
We expand $\left( {{y}^{2}}-{{z}^{2}} \right)$ using the identity $\left( {{a}^{2}}-{{b}^{2}} \right)=\left( a+b \right)\left( a-b \right)$ ,
$\Rightarrow -{{x}^{2}}\left( y-z \right)+x\left( y-z \right)\left( y+z \right)-yz\left( y-z \right)$
Taking$\left( y-z \right)$ from whole expression,
$\begin{align}
  & \Rightarrow \left( y-z \right)\left[ -{{x}^{2}}+x\left( y+z \right)-yz \right] \\
 & \Rightarrow \left( y-z \right)\left[ -{{x}^{2}}+xy+xz-yz \right] \\
\end{align}$
Arrange the term in brackets in descending order of the power of y.
$\begin{align}
  & \Rightarrow \left( y-z \right)\left[ xy-yz-{{x}^{2}}+xz \right] \\
 & \Rightarrow \left( y-z \right)\left[ -y(-x+z)+x(-x+z) \right] \\
 & \Rightarrow \left( x-y \right)\left( y-z \right)\left( z-x \right) \\
\end{align}$
Hence, the factors of the expression $x\left( {{y}^{2}}-{{z}^{2}} \right)+y\left( {{z}^{2}}-{{x}^{2}} \right)+z({{x}^{2}}-{{y}^{2}})$ are $\left( x-y \right)\left( y-z \right)\left( z-x \right)$
Note: The key concept involved in solving this problem is the knowledge of factors involved in an expression. In such problems, students must know how to rearrange different terms so that they can extract common terms from the whole expression and reduce a complex expression into a simpler and easier expression. This type of problem requires much practice so that student gets familiar with the topic.