Question

How do you factor `25c ^ { 2} - 40cd - 64d ^ { 2}`?

Answer 1

You can't

Explanation:

Just by eyeballing the expression, and maybe with a little experience, you may notice that the expression looks like the square of a binomial:

`(a\pmb)^2 = a^2\pm2ab+b^2`

In fact, there are three terms, and the first and the last are perfect squares.

Since `25c^2 = (5c)^2 = a^2` and `64d^2 = (8d)^2 = b^2`, this expression is the square of a binomial if and only if the remaning term `-40cd` is, sign apart, twice the multiplication of the other two terms `2ab`.

Unfortunately, you can see that `ab` is already `40cd`. This means that this expression can be written as

`a^2+ab+b^2`

This expression cannot be factorized, i.e. cannot be write as a multiplication of two (or more) polynomial of smaller degree.

Note that writing something like

`a^2+ab+b^2 = a(a+b)+b^2` is not a factorization, since you're not writing your expression as a multiplication of other factors.