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???
for the second question i suggest that you use the rational root theorem to find a rational root, lets call it $\displaystyle r$. we then have
$\displaystyle 3x^2-2x^2+x+7=(x-r)Q(x)$ where $\displaystyle Q(x)$ is a quadratic.
however if there is not rational root then this becomes quite difficult.
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