# More Finding Zeros of a Polynomial

• Feb 21st 2010, 07:26 PM
JennyFlowers
More Finding Zeros of a Polynomial
Hello, all. I have three problems related to finding zeros of a polynomial.

First problem:

I need to find the zeros of the following polynomial so that I can graph it:

$x^4-2x^3-5x^2+8x+4$

Using synthetic division, I found two zeros and so now I have:

$(x+2)(x-2)(x^2-2x-1)$

I'm not sure how to factor $x^2-2x-1$ to find the remaining zeros.

Second problem:

I need to find the zeros of the following polynomial so that I can graph it:

$2x^3+6x^2+5x+2$

Using synthetic division, I found one zero, so now I have:

$(x+2)(x^2+x+\frac{1}{2})$

I'm pretty sure that $x^2+x+\frac{1}{2}$ can't be factored, so what does this mean for the graph?

Final problem:

I need to find the zeros of the following polynomial so that I can graph it:

$x^4-4x^3-x^2+14x+10$

I used synthetic division to find:

$(x+1)^2(x^2-6x+10)$

Again, it doesn't seem that I can factor any further. What does this mean for the graph?

Thanks again!
• Feb 22nd 2010, 01:36 AM
HallsofIvy
Quote:

Originally Posted by JennyFlowers
Hello, all. I have three problems related to finding zeros of a polynomial.

First problem:

I need to find the zeros of the following polynomial so that I can graph it:

$x^4-2x^3-5x^2+8x+4$

Using synthetic division, I found two zeros and so now I have:

$(x+2)(x-2)(x^2-2x-1)$

I'm not sure how to factor $x^2-2x-1$ to find the remaining zeros.

Factoring is not the only way to solve a quadratic equation. Did you try the quadratic formula?

Quote:

Second problem:

I need to find the zeros of the following polynomial so that I can graph it:

$2x^3+6x^2+5x+2$

Using synthetic division, I found one zero, so now I have:

$(x+2)(x^2+x+\frac{1}{2})$

I'm pretty sure that $x^2+x+\frac{1}{2}$ can't be factored, so what does this mean for the graph?
Again, use the quadratic formula. If the roots are complex (discriminant is negative) then the graph only crosses the x-axis at x= -2.

Quote:

Final problem:

I need to find the zeros of the following polynomial so that I can graph it:

$x^4-4x^3-x^2+14x+10$

I used synthetic division to find:

$(x+1)^2(x^2-6x+10)$

Again, it doesn't seem that I can factor any further. What does this mean for the graph?

Thanks again!
Again, use the quadratic formula. If the roots are complex (discriminant is negative) then the graph only touches (but does not cross) the x-axis at x= -1.