In Exercises 51–54, graphs of fifth-degree polynomial functions a...

Question

In Exercises 51–54, graphs of fifth-degree polynomial functions are shown. In each case, specify the number of real zeros and the number of imaginary zeros. Indicate whether there are any real zeros with multiplicity other than 1.

Graph of a fifth-degree polynomial function showing real and imaginary zeros.

Accepted Answer

Hello everyone in this video, we're going to be looking at this graph of 1/5 degree polynomial. And we want to try to determine the number of real zeros and the number of imaginary zeros. So because we have this fifth degree polynomial, so I'm just going to say X to the fifth polynomial each of these access. If we write out X to the 5th as X times X times X times X times X. Each of these exes is going to have of value, which means that we actually have five zeros. Because each of these exes is going to pass through Y equals zero. So at y equals zero. Each one of these five Xs has a value. So that represents five zero's total. So you can figure out the number of total zeros just by looking at the degree of an equation or a polynomial. In this case we have X to the fifth or 1/5 degree polynomial. So we have five zeroes and recall that real zeros are when we actually see the function across from the negative Y axis into the positive Y axis. Or essentially the function needs to pass through the X axis or cut through the X axis. So let's count the number of times that we see the function cut through the X axis. So this first time that we see it cut through is on the left hand side of the y axis, close to negative 10. And then we see it cut through the y axis here. But we actually don't count that when we're looking for zeros, we wanted to cross the X axis. So the second time that that happens is here close to 10. And then it happens for one last time, close to 20. So you can see that there's three times where we have the function cutting through the X axis. So that means that we have three real zeros. And because we already determined that we should have five zero's total, that leaves two imaginary zeroes. So you can see that just by looking at the graph and the degree of the polynomial, we were able to determine the real and the imaginary zeros. And something to note here is that imaginary zeros come in pairs? So whenever you determine the number of imaginary zeros, it should always be an even number. So this becomes the final answer for this problem. And if we look at the answer choices, we want three real zeros and two imaginary zeros. So that leaves answer choice C. So hopefully this video helped and see you next time

In Exercises 51–54, graphs of fifth-degree polynomial functions are shown. In each case, specify the number of real zeros and the number of imaginary zeros. Indicate whether there are any real zeros with multiplicity other than 1.

Key Concepts

Polynomial Functions

Real and Imaginary Zeros

Multiplicity of Zeros