Question

Find all values of k so that the trinomial x^2 + kx - 35 can be factored using integers

Answer 1

`(x-r_1)(x-r_2)=x^2-(r_1+r_2)x+r_1r_2`

So the trinomial
`x^2+kx-35` can be factored as long as


`\begin{cases}r_1+r_2=-k\\r_1r_2=-35\end{cases}`

has integer solutions for
`r_1,r_2`. Clearly, both have to be factors of -35, which leaves only a handful of cases:


`(r_1,r_2)=(1,-35)\implies r_1+r_2=-34`

`(r_1,r_2)=(-1,35)\implies r_1+r_2=34`

`(r_1,r_2)=(5,-7)\implies r_1+r_2=-2`

`(r_1,r_2)=(-5,7)\implies r_1+r_2=2`

So the possible values of
`k` are
`\pm34` and
`\pm2`.