Question

Factor r^3 + 3r^2 -10r -24

Answer 1

\[r^3 + 3r^2 - 10r - 24\]

Factored out the common factor of the polynomial:
\[(r^2 + 4 - 12)(r+2) \]

Factored the quadratic into a binomial (answer):
\[(r−3)(r+4)(r+2)\]

Answer 2

Can there also be different forms of this, like other factors?

Answer 3

And I am not sure what you mean by the common factor sorry...

Answer 4

Okay, first of all I made a typo in the second equation

the 4 is suppose to be an r

Answer 5

\[r^3 + 3r^2 - 10r - 24\]

Answer 6

I think you were missing a factor. (r-1)(r+2)(r+4)(r-3)

Answer 7

The first thing to do is find the common factor, do you know what greatest common factor is?

Answer 8

I am just looking for some insight of where to start on these. My professor told me that when the coefficients add up to zero I can use r-1 as a factor

Answer 9

Yes

Answer 10

But i dont see how r-2 goes into all those terms. I have never really looked for gcf in this way before unfortunately.

Answer 11

Okay, well in this equation first we want to find the factors of 24, and see which factor makes the equation equal to 0 when we substitute it for r.

Answer 12

ahh ok

Answer 13

So all the solutions are going to be factors of the last term?

Answer 14

What we should be left with is a factor beside a quadratic which can be simplified to 2 binomials. The first 2 expressions in the parenthesis is the quadratic simplified, and the third is actually the greatest common factor of the original equation.

Answer 15

Here's how I got the (r+2):

Answer 16

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24

Answer 17

We try to see if when we input the positive or negative value of a factor if the equation equals 0

Answer 18

And we find that when r = -2 is when the equation is equal to 0

Answer 19

And I found that by testing 1, then -1, then 2, then -2, etc up the positive and negatives of the factors

Answer 20

Basically because -2 made the equation equal to 0, (r+2) is a factor

Answer 21

Thanks this really helped a lot (;

Answer 22

No problem lol

Answer 23

Applying the rational root theorem, r = -2 is a solution to the given expression and r + 2 = 0 is a factor.

Next split given expression in accordance with factor x + 2 and factor completely:

r^3 + 3r^2 -10r -24 = r^3 + 2r^2 + r^2 + 2r - 12r - 24
= r^2(r + 2) + r(r + 2) - 12(r + 2)
= (r + 2)(r^2 + r - 12)
= (r + 2)(r + 4)(r - 3)