Question

Find two values for k such that the trinomial x2 – 7x + k can be factored over the integers. Explain your reasoning.

Answer 1

1) For
`a = 1`:
`b = 6` and
`k = 6`, 2) For
`a = 3`:
`b = 4` and
`k = 12`

Explanation:

The polynomial
`y = x^(2) - 7\cdot x + k` is a second-order polynomial of the form
`(x-a)\cdot (x-b) = x^(2)-(a+b)\cdot x + a\cdot b`. By direct comparison, we construct the following system of equations:

`a + b = 7` (1)

`a\cdot b = k` (2)

By (1) we know that there are a family of pairs such that the system of equations is satisfied. Let suppose that both
`a` and
`b` are integers. We assume two arbitrary integers for
`a`:

1)
`a = 1`

`b = 7 - a`

`b = 6`

`a\cdot b = k`

`k = (6)\cdot (1)`

`k = 6`

2)
`a = 3`

`b = 7 - a`

`b = 4`

`a\cdot b = k`

`k = (3)\cdot (4)`

`k = 12`