Thread: Finding the remaining zeros of each polynomial function

1. Finding the remaining zeros of each polynomial function

Use the given zero to find the remaining zero's of each polynomial function.

P(x)= x^3+3x^2+x+3; -i

I'm pretty lost with this stuff. I've been watching Patrick's videos trying to remember my high school days but wow it's been a while hehe.

It looks like Conjugate Pair Theorem would be the way to solve. If I figure it out I'll post my solution.

If you guys could give me some direction to help me get back into the game that would be AWESOME!!

Thanks a TON

2. Originally Posted by RBlax
Use the given zero to find the remaining zero's of each polynomial function.

P(x)= x^3+3x^2+x+3; -i

I'm pretty lost with this stuff. I've been watching Patrick's videos trying to remember my high school days but wow it's been a while hehe.

If you guys could give me some direction to help me get back into the game that would be AWESOME!!

Thanks a TON
1. Since there are only real coefficients in the term of the function there must exist a 2nd solution x = +i. Thus $\displaystyle (x^2+1)$ must be a factor in the term:

2. $\displaystyle P(x)= x^3+3x^2+x+3 = x^2(x+3) + (x+3) = (x+3)(x^2+1)$

3. A product equals zero if one factor equals zero. Use this property to determine the last missing zero of P.

3. Hello, RBlax!

Use the given zero to find the remaining zeros.

. . $\displaystyle P(x)\:=\: x^3+3x^2+x+3.\quad x = -i$

Since $\displaystyle x = -i$ is a zero of $\displaystyle P(x)$, then $\displaystyle x = +i$ is also a zero.
. .
(Complex roots always appear in conjugate pairs.)

Then: .$\displaystyle [x - (-i)]\text{ and }[x - (+i)]$ are factors of $\displaystyle P(x).$

. . That is: .$\displaystyle (x + i)(x - i) \:=\:x^2+1$ is a factor of $\displaystyle P(x).$

Dividing, we find that: .$\displaystyle P(x) \:=\:(x^2+1)(x+3)$

Therefore, the zeros of $\displaystyle P(x)$ are: .$\displaystyle i,\:-i,\:-3$

Edit: Too slow . . . again!
.

4. This isn't using the The Conjugate Pair Theorem though right?

I think what they are wanting me to do looks something like...

$\displaystyle P(x)= x^3+3x^2+x+3; -i$

then use synthetic division (which I begin to become lost)

Edit: I'm slow too haha

But thanks a ton for the help.

given that 4 is a zero of the polynomial function f(x) find the remaining zeros.

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