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tia_tia [17]
2 years ago
12

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

Mathematics
1 answer:
Korvikt [17]2 years ago
8 0
(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.
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