Question

How do i do this? The polynomial x3 - 4x2 - 29x - 24 has a factor of x - 8. Find the remaining factors. thank you in advance!

`x^3-4x^2-29x-24` has a factor of `x-8`

Answer 1

`(x-8)(x+3)(x+1)`

Explanation:

`(x-8)xx(?)=x^3-4x^2-29xcolor(green)(-24) = 0`

All the potential factors of 24 are

`+-1;+-24;+-2;+-12;+-3;+-8;+-4;+-6`

We must have 3 of them if there are 3 factors in the solution

The factors are the x-intercept. So a cubic has just 1 or 2 or 3 factors. The 2 factor scenario is where the x-axis is tangential to a local minimum

We are already given `(x-8)=0 =>color(red)(x=+8)`
Lets investigate for 3 factors.

We know that we have `(x-8)` and the constant `color(green)(-24)` means that we need to build `(-24)/(-8)=+3`

As 3 is a prime number we have the option of `+1 and +3` as values of constants.

Testing:
`(x-8)(x+3)(x+1)`

`(x^2-5x-24)(x+1)`

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

`x^3-4x^2 -29x-24 color(red)(larr" Works")`

By the way; notice that `-8+3+1=-4` matching `-4x^2`

`color(magenta)("So the product must equal the constant and the sum must equal")` `color(magenta)("the coefficient of "x^2)`