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:

$U=5\,x$

$V=6\,y^4$

Step-by-step 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:

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