Question

We want to factor the following expression: 25x^2−36y^8 We can factor the expression as (U+V)(U-V) where U and V are either constant integers or single-variable expressions. What are U and V?

Answer 1

Answer: U= 5x*3 and V=3

Factored: (5x^3 -3)^2

Answer 2

`U=5\,x`

`V=6\,y^4`

Explanation:

Recall that an expression that can be factored as (U+V)(U-V) using distributive property for multiplication of binomials, should render:
`U^2-V^2` (the factorization given above is that of a difference of squares. Then, the idea is to write the original expression :

`25\,x^2-36\,y^8`

as a difference of perfect squares. Let's examine each term and its numerical and variable form to find if they can be written as perfect squares:

a) the term
`25\,x^2=5^2\,x^2=(5x)^2` therefore, if we assign the letter U to
`5\,x`, the first term becomes:

`25\,x^2=(5\,x)^2=U^2`

b) the term
`-36\,y^8=-6^2\,(y^4)^2=-(6\,y^4)^2` therefore, if we assign the letter V to
`6\,y^4` , this second term becomes:

`-36\,y^8=-(6\,y^4)^2=-V^2`

With the above identification, our expression can now be factored as a difference of squares:

`25\,x^2-36\,y^8=(5\,x)^2-(6\,y^4)^2=U^2-V^2=(U+V)(U-V)`