Given:
The expanded form of [tex](x+s)(x-t)[/tex] is [tex]x^2+Kx-40[/tex].
Where, s, t, K are positive integers.
To find:
The smallest possible value of K.
Solution:
The expanded form of [tex](x+s)(x-t)[/tex] is [tex]x^2+Kx-40[/tex]. It means,
[tex](x+s)(x-t)=x^2+Kx-40[/tex]
[tex]x^2-tx+sx-st=x^2+Kx-40[/tex]
[tex]x^2+(s-t)x-st=x^2+Kx-40[/tex]
On comping both sides, we get
[tex]K=s-t[/tex] ...(i)
K is a positive integer if s>t.
And
[tex]st=40[/tex]
The factor pairs of 40 are (1,40), (2,20), (4,10), (5,8), (8,5), (10,4), (20,2) and (40,1).
Since s>t, therefore the possible values for (s,t) are (8,5), (10,4), (20,2) and (40,1).
Using (i), find the value of K for each factor pair.
[tex]K=8-5=3[/tex]
[tex]K=10-4=6[/tex]
[tex]K=20-2=18[/tex]
[tex]K=40-1=39[/tex]
Therefore, the smallest possible value of K is 3.
