Question

How do you find all values of k so that the polynomial `x^2+kx+14` can be factored with integers?

Answer 1

`k in {-15,-9,9,15}`

Explanation:

In order to factorise this quadratic with integers we would require factors of `14` that add up to `k`. We can just list these:

`{: ("factor1", "factor2", "sum", "factored quadratic") , (14,1,15,(x+14)(x+1)), (7,2,9,(x+7)(x+2)), (-14,-1,-15,(x-14)(x-1)), (-7,-2,-9,(x-7)(x-2)) :}`

So the values are `k in {-15,-9,9,15}`

Answer 2

`k in {-15,-9,9,15}`

Explanation:

Solving `x^2+k x+14=0` we have

`x=1/2(-kpmsqrt(k^2-56))`. We need

`k^2-50= m^2` where `m, k` are integers

so

`k^2-m^2=56`

Here `56 = 1 cdot 2^3 cdot 7`

so the possible factors are

`f = {1,2,4,7,8,14,28,56}`.

The solutions for `m,k` are the integer solutions for

`{(k+m=f_i),(k-m=56/f_i),(-k+m=f_i),(-k-m=56/f_i):}`

The positive solutions kor `k` are

`((k,m),(9,-5),(15,-13),(9,5),(15,13))`

Finally, the feasible values for `k` are `{-15,-9,9,15}`

Note that

`x = 1/2(-kpmm)` and `-kpm m` is an even number so is divisible by `2`.

The factorizations are

`(x+2)(x+7)` and
`(x+1)(x+14)`
`(x-2)(x-7)` and
`(x-1)(x-14)`