Question

How do you factor `k^2-13k+40`?

Answer 1

We can Split the Middle Term of this expression to factorise it.

In this technique, if we have to factorise an expression like `ak^2 + bk + c`, we need to think of 2 numbers such that:
`N_1*N_2 = a*c = 1*40 = 40`
AND
`N_1 +N_2 = b = -13`
After trying out a few numbers we get `N_1 = -8` and `N_2 =-5`
`-8*-5 = 40`, and `-8+(-5)= -13`

`k^2-13k+40` = `k^2-8k-5k+40`

`= k(k-8) - 5(k-8)`

`(k-8)` is a common factor to each of the terms

`=color(green)((k-8)(k-5)`

Answer 2

There is another way that avoids the lengthy factoring by grouping.
`f(x) = x^2 - 13x + 40 =` (x - p)(x - q).
Find 2 numbers p and q knowing product c = 40 and sum b = -13.
Compose factor pairs of (40). Proceed: (2, 20)(4, 10)(5, 8). This last sum is 13 = -b.
Then p = -5 and q = -8

f(x) = (x - 5)(x - 8).

No need to factor by grouping.