Question

question #8 the graph of f(x) = 1/2(x³ + 3x² - 4x - 12) is shown below. determine the number of imaginary solutions of f(x). 0 2 1 3

Explanation:

Step1: Recall fundamental theorem of algebra

A polynomial of degree $n$ has $n$ roots (counting multiplicities) in the complex - number system. The given function $f(x)=\frac{1}{2}(x^{3}+3x^{2}-4x - 12)$ is a cubic polynomial, so $n = 3$.

Step2: Determine real roots from the graph

The real roots of the function $y = f(x)$ are the $x$ - values where the graph of the function intersects the $x$ - axis. From the graph, we can see that the graph of $y=f(x)$ intersects the $x$ - axis at 3 distinct points. So, the number of real roots is 3.

Step3: Calculate the number of imaginary roots

Let the number of real roots be $r$ and the number of imaginary roots be $i$. Since the total number of roots of a cubic polynomial is 3, and complex (non - real) roots come in conjugate pairs. We know that $r + i=3$. Since $r = 3$, then $i=0$.

Answer:

0