Question

Find all the zeros of each equation.
x^5 – 3x^4 – 15x^3 + 45x^2 – 16x + 48 = 0

Answer 1

Do you know how to find the possible rational roots?

Answer 2

No, I'm lost.

Answer 3

List out the factors of 48 (the last number) and the factors of 1 (the first coefficient)


*** List both the positive and negative factors ***


Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48, -1, -2, -3, -4, -6, -8, -12, -16, -24, -48


Factors of 1: 1, -1



Now divide each factor of 48 by each factor of 1 (there are 96 divisions going on here, but they all lead to this list)


1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 8, -8, 12, -12, 16, -16, 24, -24, 48, -48

Answer 4

oh made a typo, didn't mean to say 96 divisions, meant to say 20 divisions...but that's irrelevant

Answer 5

anyways, the possible rational roots are

1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 8, -8, 12, -12, 16, -16, 24, -24, 48, -48

Answer 6

Now, to find the actual zeros, you plug in each value into x^5-3x^4-15x^3+45x^2-16x+48 and see which value gives you zero.


For example, if x = 1, then


x^5-3x^4-15x^3+45x^2-16x+48


(1)^5-3(1)^4-15(1)^3+45(1)^2-16(1)+48 = 60


Since the result is NOT 0, x = 1 is NOT a root for x^5-3x^4-15x^3+45x^2-16x+48

------------------------------------------------------------

However, if x = 3, then:


x^5-3x^4-15x^3+45x^2-16x+48


(3)^5-3(3)^4-15(3)^3+45(3)^2-16(3)+48 = 0


Since the result is 0, x = 3 is a root for x^5-3x^4-15x^3+45x^2-16x+48

Answer 7

So this at least narrows down the possible list of choices

Answer 8

Does that make sense?

Answer 9

Yes, thanks.

Answer 10

yw

Answer 11

tell me what roots you get

Answer 12

4

Answer 13

4 is another root, good

there's one more in this list

Answer 14

-4

Answer 15

nice work

so the 3 roots are: 3, 4, -4

Answer 16

This is a 5th degree polynomial, so there are 5 roots total.

This means that there are 2 more roots we've yet to find.

Answer 17

To find the 2 missing roots, we use either polynomial long division or synthetic division

Answer 18

is the other roots i and -i?

Answer 19

you are 100% correct

so the 5 roots are: 3, 4, -4, i, -i

Answer 20

Okay, thanks! (:

Answer 21

yw