In Exercises 16–17, find the zeros for each polynomial function a...

Question

In Exercises 16–17, find the zeros for each polynomial function and give the multiplicity of each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x) = -2(x - 1)(x + 2)^2(x+5)^2

Accepted Answer

Hi there. So for this problem says fourth given polynomial buying the zeros. And for east zero say its multiplicity and comment it crosses the X axis or touches the X axis and turns around. So let's go and look at our polynomial. We have negative four times X plus one X minus three Q times x minus two to the fourth. So let's go and get out. Our factors are factors will be X plus one, x minus three and X minus two. Now to find our zeros we should set each of these equal to zero and solve for X. Some would say X plus one equals zero, X minus three equals zero and x minus two equals zero. To solve C. For this one. I would move the one by subtracting one meaning I get X equals negative one. This next one we move our three. Let me get x equals three. The last one we move our two And then we get x equals two. So we have our three zeros negative 13 and two. Now I'm gonna check this multiplicity multiplicity is given by the degree of the factor in our polynomial. So good luck here. X plus one doesn't have a number up here. That implies that this is just the first power Meaning that or x equals native one. Multiplicity is going to be equal to just one. Now check X equals three. X equals three. We look at our polynomial here, we have a three there. This multiplicity. It's gonna be equal soon. Finally get the last one X -2 Or x equals two. That multiplicity is four So that's equal to four. Now with multiplicity we can tell if it crosses the X. Axis or touches the X. Axis it crosses the X axis. If the multiplicity is an odd number and it goes through the excess or turns around if the multiplicity is an even number. So if you like here multiplicity of one is odd, one is odd. That means it's gonna cross Well pussy of three, it's odd. So it's also going to cross At four which is even it's gonna turn around. So if you look at our answers we're gonna look at which one fits all these criteria. All of our zeroes are the same in all ways except for D. So D. Is out. And if you let me have multiplicity of 13 and four. So we can check the crossing the turning, we look at our answers. C. Is the answer that has all of the things we found in this problem which means that our answer is answer. C. Okay I hope to help you solve the problem. Thank you for watching. Goodbye

In Exercises 16–17, find the zeros for each polynomial function and give the multiplicity of each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x) = -2(x - 1)(x + 2)^2(x+5)^2

Key Concepts

Zeros of a Polynomial

Multiplicity of Zeros

Graph Behavior at Zeros