Question

What are the factors of x^3 + 3x^2 - 10x - 24?

Answer 1

I know one of the factors are (x+3), but can someone show me step by step how to factor this?

Answer 2

@ganeshie8 can you help me please?

Answer 3

(x-3)(x+2)(x+4)

Answer 4

Have you been taught the rational roots theorem yet?

Answer 5

can you show me the step by step procedure?

Answer 6

i have been taught that. I know how to do this, it is that i forgot how to...

Answer 7

How did you know one of the factors is (x+3)?

Answer 8

well actually it is (x-3) and i was trying to factor it by grouping but i did it wrong..

Answer 9

Okay I still don't know how you figured (x-3) is a factor (which happens to be correct). Do you want to know how to solve this problem from scratch or do you just want to know how to proceed after knowing (x-3) is a factor?

Answer 10

from scratch please

Answer 11

This is a cubic equation and so there will be three roots.

You start by testing a few small values of x. If you want to get into rational roots theorem we can go that way. But if you don't want that route you can try a few small values of x such as 1, -1, 2, -2, 3, -3, etc and until you find an x value that makes
f(x) = x^3 + 3x^2 - 10x - 24 equal zero.
What is f(1), f(-1), f(2), f(-2)?

Answer 12

can we use the rational root theorem, i think that is when you look at the multiplies of the first and last terms right?

Answer 13

Yes. Take the absolute value of the constant term (24, don't worry about the sign) and the leading coefficient (1). The possible roots are: \[\Large \pm\frac{ \text{ all factors of 24} }{ all factors of 1 } = \pm \text{ all factors of 24} = \pm(1,2,3,4,6,8,12,24)\]

Answer 14

Possible roots are: +/- (1, 2, 3, 4, 6, 8, 12, 24)

So you need to try x = 1, -1, 2, -2, 3, -3, etc. But you can stop after finding just one root and then we can go to the next step.

Answer 15

What is f(1), f(-1), f(2), f(-2) ?

Answer 16

1 and -1 won't work but coould you show me using synthetic division to solve for 2 and -2. I feel like i am doing something wrong while doing it.

Answer 17

You don't do synthetic division yet. You have to find one root first.
f(x) = x^3 + 3x^2 - 10x - 24
f(2) = 2^3 + 3(2)^2 -10(2) - 24
= 8 + 12 - 20 - 24
= -24

Try x = -2
f(-2) = (-2)^3 + 3(-2)^2 -10(-2) - 24
= -8 + 12 + 20 - 24
= 0

Therefore, -2 is a root which implies (x + 2) is one factor which means:
x^3 + 3x^2 - 10x - 24 = (x + 2)( some lower degree polynomial )
To find the lower degree polynomial divide: (x^3 + 3x^2 - 10x - 24) / (x + 2)

This is where you use synthetic division.

Answer 18

ohhh i get it, so the answers would be x+2, x-3 , x+4?

Answer 19

Yes, that will be the final answer. But you have to do synthetic division first.

Answer 20

-2 | 1 3 -10 -24
| -2 -2 24
|__________________________

1 1 -12 0

The quotient is: x^2 + x -12

x^3 + 3x^2 - 10x - 24 = (x + 2)(x^2 + x -12)

You can factor the quadratic expression: (x^2 + x -12)
and then you will have all three factors.