Question

How do you factor \[{{x}^{3}}-3{{x}^{2}}=0\]?

Answer 1

Hint:
First, understand the meaning of the term ‘factoring’ and then follow the steps to factorize the given polynomial. Take the common terms together and write the remaining terms inside the bracket to get the factored form of the provided expression.

Complete step by step answer:
Here, we have been provided with the polynomial (cubic equation) \[{{x}^{3}}-3{{x}^{2}}=0\] and we are asked to factorize this polynomial. But first, let us see the meaning of the term ‘factoring’.
Now, in mathematics, factorization or factoring is a method of writing a number or polynomial expression as a product of several factors. For example: - let us consider a number 10, so we can write 10 as: - \[10=2\times 5\]. Here, 2 and 5 are called factors of 10, and the process is known as factorization. Let us take an example of a polynomial expression given as: - \[{{x}^{2}}-9\]. The considered expression can be written as: -
\[\Rightarrow {{x}^{2}}-9={{x}^{2}}-{{3}^{2}}\]
Now, using the identity, \[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\], we get,
\[\Rightarrow {{x}^{2}}-9=\left( x+3 \right)\left( x-3 \right)\]
Here, \[\left( x+3 \right)\] and \[\left( x-3 \right)\] are called factors of the polynomial and the process by which these factors are found is called factorization.
Now, let us come to the question. We have the polynomial: \[{{x}^{3}}-3{{x}^{2}}=0\]. Here, we will use the grouping method to get our answer. What we will do is we will group the common terms together and write the remaining terms in brackets. So, we have,
\[\Rightarrow {{x}^{3}}-3{{x}^{2}}=0\]
The above expression can be written as: -
\[\Rightarrow {{x}^{2}}\times x-3{{x}^{2}}=0\]
We can see that \[{{x}^{2}}\] is common in both the terms, so taking \[{{x}^{2}}\] common, we get,
\[\Rightarrow {{x}^{2}}\left( x-3 \right)=0\]
\[\Rightarrow x\times x\times \left( x-3 \right)=0\] - (1)
So, there are a total of three factors for the given polynomial. Two of the factors are the same i.e. x, and the third factor is (x – 3). Hence, equation (1) represents the given expression in factored form.

Note:
 One may note that this was a cubic equation and that is why we obtained three factors. So, the highest power of the variable determines the number of factors of the given polynomial. Now, if we would have been asked to determine the roots of the expression then we would have substituted each term of the factored form equal to 0. You may see that in the provided expression the coefficient of ‘x’ and the constant term is 0. If the polynomial were of the form \[a{{x}^{3}}+b{{x}^{2}}+cx+d\] where (a, b, c, d) \[\ne \] 0 then, in that case, we have to find one factor by hit and trial method and the remaining two by the middle term split method, to get the factored form.