# f(x) and g(x) compositions

• Dec 7th 2006, 07:07 PM
Mr_Green
f(x) and g(x) compositions
f(x) = 3x^3 - 2x^2 + x + 7

g(x) = 2x^2-4

2 QUESTIONS:

Find g of f(x)

AND...

f(x) has two complex zeros. One is in the 4th quadrant. In what quadrant is the other???
• Dec 7th 2006, 07:10 PM
Ruichan
g of f(x)
= 2* (3x^3 - 2x^2 + x + 7)^2 - 4
• Dec 7th 2006, 08:30 PM
putnam120
for the second question i suggest that you use the rational root theorem to find a rational root, lets call it $r$. we then have

$3x^2-2x^2+x+7=(x-r)Q(x)$ where $Q(x)$ is a quadratic.

however if there is not rational root then this becomes quite difficult.
• Dec 8th 2006, 04:32 AM
topsquark
Quote:

Originally Posted by Mr_Green
f(x) has two complex zeros. One is in the 4th quadrant. In what quadrant is the other???

Two observations:
The question states there are only two complex zeros.
f(x) is a cubic polynomial so there can be at most two complex zeros.

Either way you look at it, there are ONLY two complex zeros of f(x). Since f(x) is a polynomial with real coefficients the two complex zeros are complex conjugates of each other.

Since one complex zero is in the 4th quadrant, it must have the form a - bi, where a, b > 0. Since the other zero is the complex conjugate of this, the second zero is of the form a + bi, which would be in the first quadrant.

-Dan