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

2 Answers
May 19, 2015

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)#

May 19, 2015

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.