Question

can someone help me with Nature of roots and zeros?
Find all the zeros of each polynomial function.

f(x) = x4 – 5x2 – 36


1) Select one:
a. ±3, ±2i
b. ±3 only
c. ±2i only
d. none of the above

To find the roots of a polynomial equation you should:

Select one:
a. only use Descartes Rule of Signs.
b. only use the Quadratic Formula.
c. only factor it.
d. use any method that gives the roots of the function

Answer 1

Nikki: One option available to you would be substituting x^2=y. If you do this, you end up with y^2 - 5y -36. What are the roots of this quadratic equation? Write out all that apply.

Answer 2

For second I think d is correct

Answer 3

Depends on question which method suits it.

Answer 4

yeah I say D too

Answer 5

@mathmale the root would be 2 right?

Answer 6

Plug it into the equation then if you get its value =0 then its correct

Answer 7

i dont know how to?

Answer 8

the function you have can be factored

\[x^4 -5x^2 - 36 = (x^2 - 9)(x^2 + 4)\]

now you can solve each factor

\[x^2 - 9 = 0~~~~~~~~~~~~~~ and~~~~~~~~~~~x^2 + 4 = 0\]

Answer 9

the equation is called an equation that is reducible to a quadratic.... you can make a substitution as some people have suggested...and then factor , or simply factor...

hope it helps

Answer 10

so if there both 0 its correct but then how do you know?

Answer 11

well to find roots or zeros... then y or f(x) or P(x) needs to be equal to zero...

what you have written in the question is simply an algebraic expression that can be factored...

Answer 12

huh?

Answer 13

wait the roots for both of those is 2?

Answer 14

well can you solve

\[x^2 - 9 = 0\]

that is one factor... so if that factor equals zero, the whole equation equals zero...

so solve for x.

the other factor is

\[x^2 + 4 = 0\]

same thing, find the values of x that make the factor zero...

remember the question asks you to find the values of x, often called roots or zeros that make the equation equal zero.

Answer 15

-2i?

Answer 16

well remember you are solving a degree 2 equation so 2 solutions...

Answer 17

-3 is the other one

Answer 18

The original equation is of the fourth order (the highest power is 4). Thus, there should be FOUR solutions.

Answer 19

how do you get 4 solutions?

Answer 20

If the highest x power in the given equation is n, then the order of the polynomial is "n th." Thus, because the highest power of x that you're working with here is 4, it's a fourth-order polynomial and has 4 solutions.

Answer 21

x^2 - 9 = 0 is a 2nd order poly and thus has 2 solutions.

Answer 22

how do i find 4 solutions if theres only 2 mini equations to work with?

Answer 23

Please review campbell_st's first comment. He was right on target there. He factored the original function into two factors, each of which has 2 factors. 2 * 2 = 4 factors total.

Answer 24

Each "mini-equation" has 2 factors and thus two solutions.

Answer 25

You've already tried to factor and then solve x^2+4=0.
You got -2i as one root. The other is +2i. We call these roots "complex pairs."

Answer 26

and the other one is -3 and +3?

Answer 27

Yes, provided that you began with x^2-9=0.

Answer 28

I'm assuming you're satsified and finished with this problem. If not, please say so. Have to get off the computer!

Answer 29

so its +3 and +2 ?

Answer 30

- you've forgotten the negatives!

Answer 31

But that reply contains only 2 solutions. Just earliler, we got 4! You have to list all four solutiions.

Answer 32

List them as a set: {3i, 2, [and two more]}

Answer 33

complex roots such as 3i stay complex; do not drop the "i".

Answer 34

2i right?

Answer 35

First of all, Nikki, how many solutions does your original problem have?

Answer 36

4

Answer 37

Yes. So you MUST list all four.
If the first complex root is 3i, what is the accompanying complex root?

Answer 38

2i

Answer 39

There is no complex root containing a '2' both complex roots contain a '3'.
If the first complex root is 3i, what is the accompanying (negative) complex root?

Answer 40

complex roots just about always come in PAIRS.

Answer 41

-3i? and the other one is 2i and -2i

Answer 42

2i and -2i are the roots of x^2+4=0. The roots of x^2-9=0 are real. Again, please go back and review Campbell_st's first post. Looks like you and I have both made a few minor mistakes. Campbell is correct.

What are the two (complex) roots of x^2+4=0?
What are the two (real) roots of x^2-9=0?

The four solutions of the original equation are { you fill in }

Answer 43

huh? so 3 isnt correct?

Answer 44

Nikki, please recognize that this equation has FOUR solutions. Please don't type out just one. 3 is one of the two real roots of x^2-9=0. What is the other real root? What are the two (complex) roots of x^2+4=0?

Answer 45

Nikki: Please fill this out: The four roots of the original equation are { }.

Answer 46

I dont know

Answer 47

ive told you what i know

Answer 48

What are the two roots of x^2+4=0? We've gone over this before.

Answer 49

-2 and +2

Answer 50

Actually, 2i and -2i. So the set of your complex roots is {2i,-2i} (a pair of roots).

Answer 51

what are the roots of x^2-9=0? Write them as a set, please.

Answer 52

as a set?

Answer 53

{2i,-2i} is a SET of roots, your complex roots.

{ ? , ? } is a SET of your real roots. Please fill in.

Answer 54

If I ask for a set, list the numeric values and enclose all of the values within { } to show that they are a SET.

Answer 55

Nikki, what are the roots of x^2+4=0? Hint: there are two of them, and both are complex. Write them as a set, please. Answer: {2i, -2i}.

Answer 56

what are the roots of x^2-9=0-? There are 2 of them, and both are real. Write them as a set, please.

Answer 57

Your answer:

Answer 58

By now you are probably tired and frustrated. Do you want to finish this problem now or finish it later? Your choice.

Answer 59

Sorry, Nikki, but I'm logging off. I apologize for the extreme length of this discussion. Both of us made some mistakes, which helped to extend the discussion. I'd suggest that you either return to this discussion when you're ready, or repost it, so all discussion will be fresh. You might want to review this discussion, summarize what you've learned from it, and make up a list of questions to ask for clarification. Take care. mathmale