Question

Factorize the polynomial $45x{(x - 2)^2} + 60x(x - 2) + 20x$.

Answer 1

Hint: Factorization of polynomials expressed as a polynomial with the coefficients in the integers as the product of irreducible factors with coefficients in the same domain . For this given question we factorize this using the middle term method . In quadratic factorization using the splitting of the middle term which is $x$ term is the sum of two factors and the product of two factors is equal to the last term of the quadratic polynomial.

Complete step by step answer:
Given polynomial $45x{(x - 2)^2} + 60x(x - 2) + 20x$
Take common $x$ from the given polynomial and we have
\[ = x\{ 45{(x - 2)^2} + 60(x - 2) + 20\} \]
Split \[{(x - 2)^2}\] by using formula of ${(a - b)^2} = {a^2} - 2ab + {b^2}$ , we get
$ = x\{ 45({x^2} - 4x + 4) + 60(x - 2) + 20\} $
Multiplying and arrange the polynomial , we get
$ = x(45{x^2} - 180x + 180 + 60x - 120 + 20)$
Calculate and subtract from the above equation and we have a quadratic polynomial
$ = x(45{x^2} - 120x + 80)$ ……………………………………….(1)
Factorize the above polynomial and we get
$ = x\{ 45{x^2} - (60 + 60)x + 80\} $
In the above line when we multiply the middle part we take care about sign i.e., $( + )( - ) = ( - )$ and $( - )( - ) = ( + )$
$ = x(45{x^2} - 60x - 60x + 80)$
Take common part by part and get
$ = x\{ 15x(3x - 4) - 20(3x - 4)\} $
Take common $(3x - 4)$ from the above polynomial , we get
$ = x(3x - 4)(15x - 20)$
Take common of $5$ from the above polynomial and we have
$ = 5x{(3x - 4)^2}$
Therefore the factorization of the given polynomial is $5x{(3x - 4)^2}$.

Note:
> We can also solve the given problem by using the like , From the (1) , we calculate the answer by using formula of ${(a - b)^2}$ i.e., ${(a - b)^2} = {a^2} - 2ab + {b^2}$. Take the common of $5$ and get $(9{x^2} - 24x + 16)$ which is a square of $(3x - 4)$.
> We have the sign property, we take care about that and we need to check every line after solving the equation. In the middle term method sign property takes a huge part of this.