Question

How do you factor `2x^2-x-21`?

Answer 1

`2x^2-x-21 = (2x-7)(x+3)`

Explanation:

Assuming that `2x^2-x-21`
can be written as `(ax+-c)(bx+-d)`
`color(white)("XXXX")` `color(white)("XXXX")`with `a, b, c, d in ZZ`

Since the sign of the last term (`-21`) is negative the interior signs of the two binomials must be opposite
`color(white)("XXXX")` `rarr` we want `(ax-c)(bx+d)`;
and
since the sign of the middle term (`-x`) is negative
`cb > ad`

Looking at factors of 2: `(ab) in {(2,1)}`
and factors of 21: `(c,d) in { (3,7), (7,3), (1,21), (21,1)}`

we find
`color(white)("XXXX")` `(a,b) = (2,1)` and `(c,d) = (7,3)`
give us
`color(white)("XXXX")` `a*d - b*c = (-1)`

So our factors are
`color(white)("XXXX")` `(2x-7)(x+3)`