Question

How do you factor completely `x^2+5x+7x+35`?

Answer 1

Factor by grouping to find:

`x^2+5x+7x+35 = (x+7)(x+5)`

Explanation:

`x^2+5x+7x+35 = (x^2+5x)+(7x+35) = x(x+5)+7(x+5)`

`= (x+7)(x+5)`

With this problem, the hard work has already been done for you in the way that the middle terms are split as `5x+7x`. Normally you would be faced with something like `x^2+12x+35` and have to find the split. The trick is to find two numbers which add up to the coefficient `12` of the term in `x` and multiply together to give the constant term `35`.

Here's another example:

`x^2+14x+40`

We want to find two numbers that add up to `14` and multiply to give `40`. Having found that `10` and `4` work, we have:

`x^2+14x+40 = (x+10)(x+4)`

In general, we can say:

`(x+a)(x+b) = x^2+(a+b)x+ab`