Question

the polynomial function (f(x)=x^{6}-11x^{5}+42x^{4}-72x^{3}+71x^{2}-61x + 30) is graphed. determine the number of imaginary solutions the function has.
0
1
2
3

Explanation:

Step1: Recall fundamental theorem of algebra

A polynomial of degree $n$ has $n$ roots (counting multiplicities) in the complex - number system. The degree of the polynomial $f(x)=x^{6}-11x^{5}+42x^{4}-72x^{3}+71x^{2}-61x + 30$ is $n = 6$, so it has 6 roots in the complex - number system.

Step2: Use the relationship between real and imaginary roots

The number of real roots of a polynomial function can be determined by the number of times the graph of the function intersects the $x$ - axis. By looking at the graph of the polynomial function, we can count the number of real roots. From the graph, the polynomial function $y = f(x)$ intersects the $x$ - axis 4 times.

Step3: Calculate the number of imaginary roots

Let the number of real roots be $r$ and the number of imaginary roots be $i$. Since complex non - real roots of a polynomial with real coefficients come in conjugate pairs (if $a+bi$ is a root, then $a - bi$ is also a root), and $r + i=6$. We know $r = 4$, so $i=6 - 4=2$.

Answer:

C. 2