Why would the polynomial, "x^2+5x-24" factor out to (x-3)(x+8)?

Question

Why would the polynomial, "
`x^2+5x-24`" factor out to (x-3)(x+8)?

Answer 1

Explanation:

5x can also be written as " -3x + 8x " so:

`x^2+5x-24 = x^2-3x+8x-24`

Using the distributive property:

`x(x-3)+8(x-3)`

Then again, using the distributive property (but in reverse):

`(x-3)(x+8)`

The two numbers that the middle term should be split into in order to factor can be found by listing out pairs of factors for the product of the last term and the coefficient of the first term, and using the two factors whose sum is the middle term.

For eg (because that was a bit text heavy):

The product of the last term and the first coefficient (1 * (-24)) is -24

the factors of -24 include (and their sum)

-1, 24 (23)

1, -24 (-23)

-2, 12 (10)

2, -12 (-10)

-3, 8 (5)

3, -8 (-5)

-4, 6 (2)

4, -6 (-2)

The middle coefficient is 5 and the sum of the factors -3 and 8 are 5, so split 5x into -3x + 8x to be able to factor how I showed before.