In Exercises 25–32, find the zeros for each polynomial function a...

Question

In Exercises 25–32, find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=x^3+7x^2−4x−28

Accepted Answer

Welcome back. I'm so glad you're here. We are given this function F of x equals X cubed plus eight, X squared minus nine, X minus 72. And we're asked to find several different things. We need to identify the zeros of the given polynomial function. We need to state the zeros, multiplicity ease and we need to describe how the graph behaves with respect to the X axis. All right. In order to find those zeros, we're going to start off by factoring this function. We have four terms here. So we're going to give it a go with factoring by grouping Well, group the first two terms in the second two terms together. And we take a look at each group and ask ourselves, is there anything we can factor out for X cubed plus eight X squared. We can factor out an X squared. And when we divide both of those terms by X squared, we're left with X plus eight. For the second group here, negative nine. X minus 72. We can factor out a negative nine from both of those two terms. And when we divide both of them by negative nine, we are left with X plus eight. Which is great because both groups have a common factor this X plus eight, we can factor X plus eight out because it is being multiplied by both X squared and a negative nine. Alright, X plus eight is as factored as it's going to get but X squared minus nine. That's a difference of squares. So I'll bring down this X plus eight and we can factor our difference of squares. We recall from previous lessons that two factor a difference of squares will take the square root of both terms. The squared of x squared is X. That goes in the first spot in each set of parentheses and the square of nine is three. That goes in the second spot for each set of parentheses. And by definition one will be a plus and one will be a minus. And those are as factored as they're going to get. So we can take each factor X plus eight, X plus three and x minus three and set them all equal to zero. Now let's get our exes by themselves so that we can determine what our zeros are. So for X plus eight equals zero. That will be X equals negative eight. There's one for X plus three equals zero, that's X equals negative three. There's another zero and for x minus three equals zero. That will be an X equals positive three. There is our third zero for this polynomial function. All right, what do we do next next? We need to state their multiplicity ease. We were called from previous lessons that multiplicity. These are just telling us how many of each zero there are and four X equals negative eight. There's only one negative eight. So it has a multiplicity of one. For x equals negative three, there's only one negative three. So it has a multiplicity of one. For x equals three. There's only 13. So it has a multiplicity of one. Great, we got that one done. Now we need to describe how it behaves with respect to the X axis. Alright, when the multiplicity is odd, then the function is going to pass right through The x axis at a zero. When the multiplicity is even, then the function is going to boop the X axis at that zero, it is going to touch and then turn around when it gets to that zero. So with the multiplicity of one that's odd, A multiplicity of one. That's odd. A multiplicity of one. That's odd. So at every single zero At -8 at -3. And at positive three, the function is going to pass right through the X axis, we look at our answer choices and that matches with answer choice. A well done. We'll catch you on the next one.

In Exercises 25–32, find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=x^3+7x^2−4x−28

Key Concepts

Zeros of a Polynomial

Multiplicity of Zeros

Graph Behavior at Zeros

Watch next