Question

Factorise: $ {(5x - 3y)^2} - {(3x - 5y)^2} $

Answer 1

Hint: As we know that to multiply means to increase in number especially greatly or in multiples. Multiplicand refers to the number multiplied and multiplier refers to the number that multiplies the first number. Here in this question we have to find the product of two polynomials, we just multiply each term of the first polynomial by each term of the second polynomial and then simplify or if there is any algebraic identity possible we can apply that.

Complete step by step solution:
Here we have $ {(5x - 3y)^2} - {(3x - 5y)^2} $ ; we know the difference formula $ {(a - b)^2} = {a^2} - 2ab + {b^2} $ .
So we will first simplify the first polynomial and then the second. The first one is $ {(5x - 3y)^2} $
In this case $ a = 5x,b = 3y $ , by applying the formula and substituting the values, we get
 $ = {(5x)^2} - 2(5x)(3y) + {(3y)^2} $ $ \Rightarrow 25{x^2} - 30xy + 9{y^2} $ .
Now we will solve the second polynomial i.e. $ {(3x - 5y)^2} $
In this case $ a = 3x,b = 5y $ , by applying the formula and substituting the values, we get
 $ = {(3x)^2} - 2(3x)(5y) + {(5y)^2} $ $ \Rightarrow 9{x^2} - 30xy + 25{y^2} $ .
By putting them together we have
 $ 25{x^2} + 9{y^2} - 30xy - \left( {9{x^2} + 25{y^2} - 30xy} \right) $ .
We will break the bracket now, so the sign changes:
 $ 25{x^2} + 9{y^2} - 30xy - 9{x^2} - 25{y^2} + 30xy $ .
Putting the similar terms together we have
 $ 25{x^2} - 9{x^2} + 9{y^2} - 25{y^2} \Rightarrow 16{x^2} - 16{y^2} $ .
We take the common factor out, $ 16(x + y)(x - y) $ .
Hence the required answer is $ 16(x + y)(x - y) $ .
So, the correct answer is “ $ 16(x + y)(x - y) $ ”.

Note: We should know that the algebraic identity used in the above solution is called the square of the difference of the terms formula, which is also a binomial expression or also called the special binomial product rule. It is expanded as the subtraction of two times the product of two terms from the sum of the squares of the given terms.