Question

How do you determine if `x-5` is a factor of `2x^3-4x^2-7x-10`?

Answer 1

I think it has to do with the Factor Theorem.

Explanation:

The Factor Theorem states: When a divisor is an exact factor of `f(x)`, then the remainder after division will be 0.

ie. `color(magenta)(f(a) = 0 hArr x - a)` is a factor of `color(cyan)f(x)`

So, let `color(blue)(f(x) = 2x^3 - 4x^2 - 7x -10)`

Then, you equate `color(purple)(x-5 = 0` `color(purple)(rArr x=5)`

Next, sub in `color(red)(x=5)` into the function `color(aqua)(f(x))`

`color(brown)(f(5) = 2(5)^3 - 4(5)^2 -7(5) - 10`
`color(pink)(f(5) = 250 - 100 -35 - 10)`
`color(maroon)(f(5) = 105)`

Since `color(lavender)(f(5)!=0)`

`x-5` is not a factor of `color(orange)(2x^3 - 4x^2 - 7x -10)`

Another method will be using the Long Division method.

But I don't know how to type it properly here lol.