Zeros of Cubic Poly and Inverse Funct.

• Jan 19th 2008, 12:40 PM
toop
Zeros of Cubic Poly and Inverse Funct.
I'm trying to find all zeros by either factoring and/or using the quad. formula for:

f(x) x^3 - 2x^2 - x + 2

Next, I'm trying to show that f(g((x)) = x and g(f(x)) = x for all x:

f(x) = 2x^3 + 1 or g(x) = [cube root] (x-1/2)

Thanks in advance for any help :)

Just figured out both problems. Thanks for all the help!
• Jan 19th 2008, 12:53 PM
galactus
Quote:

I'm trying to find all zeros by either factoring and/or using the quad. formula for:

$f(x)= x^{3} - 2x^{2} - x + 2$
You could try the Rational Root Theorem. Try x=2 as a root.

That is, divide $\frac{x^{3}-2x^{2}-x+2}{x-2}$. If it reduces to a quadratic, then 2 is a factor and the quadratic is easily solved for the other two factors.
• Jan 19th 2008, 12:59 PM
toop
Awesome, I came up with 1, -1, and 2. Seems like a reasonable answer to me....thanks :)
• Jan 19th 2008, 01:22 PM
Soroban
Hello, toop!

Quote:

Find all the zeros of: . $f(x) \:x\:^3 - 2x^2 - x + 2$

Factor "by grouping": . $x^2(x-2) - (x-2) \;=\;0$

Factor: . $(x-2)(x^2-1)\;=\;0\quad\Rightarrow\quad (x-2)(x-1)(x+1)\;=\;0$

Therefore: . $x \;=\;2.\:1,\:-1$

• Jan 19th 2008, 01:24 PM
toop
Thanks again!