Question

How do you factor \[y=5{{x}^{3}}-20{{x}^{2}}+20x\] ?

Answer 1

Hint: From the given question \[y=5{{x}^{3}}-20{{x}^{2}}+20x\]. For answering this question we will use factorization. Here we have to take first \[5x\] common from all the terms then we have to use the factorization method. Factorization is the process of deriving factors of a number which divides the given number evenly. Factorization writes a number as the product of smaller numbers. Factorization is the process of reducing the bracket of a reducing the bracket of a quadratic equation, instead of expanding the bracket and converting the equation to a product of factors which cannot be reduced further. There are many methods for the factorization process. Now, we will do the given question by the method of splitting the constant and doing the sum product pattern.

Complete step by step solution:
Now considering from the question we have an expression \[y=5{{x}^{3}}-20{{x}^{2}}+20x\] for which we need to derive the factors.
Now, we have to take \[5x\] common from the given equation in each term then we will get,
\[\Rightarrow y=5x\left( {{x}^{2}}-4x+4 \right)\]
We can factor the \[{{x}^{2}}-4x+4\] by below method:
Given equation is in the form of \[a{{x}^{2}}+bx+c=0\]
First, we have to divide \[4\]into the product of the two numbers and the Sum of those two numbers must be equal to the coefficient of\[x\].
Now, \[4\]can be split into the product of the two numbers.
\[4\]can be split into product of \[-2\] and \[-2\]
Their sum is also equal to \[-4\]which is equal to the coefficient of\[x\].
\[\Rightarrow y=5x\left( {{x}^{2}}-4x+4 \right)\]
\[\Rightarrow 5x\left( {{x}^{2}}-2x-2x+4 \right)=0\]
\[\Rightarrow 5x\left( x\left( x-2 \right)-2\left( x-2 \right) \right)=0\]
 \[\Rightarrow 5x{{\left( x-2 \right)}^{2}}=0\]

Therefore \[x=0,x=2\] are the factors of \[y=5{{x}^{3}}-20{{x}^{2}}+20x\]

Note: Student should know that if the polynomial is in the form of \[a{{x}^{3}}+b{{x}^{2}}+cx+d=0\] then we can’t do this method then we have to find one root by substituting some values in it then by doing factorization we will get remaining factors. But in the given question the constant term is missing so we can take x common from the three terms and then we will get a quadratic equation then we will solve the quadratic equations we got the roots. This is the finest method to solve the above question.