Graph each polynomial function. Factor first if the polynomial is...

Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=x^3+5x^2-x-5

Accepted Answer

Hello. Today we're gonna be graphing the following function given to us. So the function is X cubed plus two X squared minus nine X minus 18. So in order to graph this function, we're going to need to find the regions of increasing and decreasing of the polynomial function. But in order to do that, we're going to need to get the zeros of X. So what we're going to need to do is we're going to need to take our polynomial function set it equal to zero and solve for the zeros of X first. So we're going to need to factor the function in order to solve for X. Since we have four terms in the given function, let's go ahead and try factoring by grouping. So we're going to group up the first two terms and the last two terms together. Then what we're going to want to do is we're going to want to factor out a common factor from each of the groups in the first group. Notice that both of the terms has a multiple of X squared. So we can go ahead and factor out an X squared from the first group and in the second group, both of the terms are multiples of nine. So we can factor out a nine from the second group. However, we want this leading coefficient to be positive after the factory. So let's go ahead and factor out a negative nine from the second group. Doing so allows us to rewrite the function as X squared times the quantity X plus two minus nine times the quantity X plus two equal to zero. Now notice that we have two instances of the group X plus two. We can combine this into a single X plus two by adding the coefficients together. And doing this allows us to rewrite the function as X squared minus nine times the quantity X plus two. Now notice that X squared minus nine is a factor binomial. By using the difference of squares, we can factor this quantity to give us X plus three times X minus three. And we're going to include the factor of X plus two because that was factored out in the beginning and we're gonna set everything equal to zero. So now what we're going to do is use our zero property to set each of the factors of X equal to zero. Then we're going to solve for X. So what we get is going to be X plus three equal to zero, X minus three equal to zero and X plus two equal to zero in the left hand equation. If we subtract three to both sides, we're going to get X is equal to negative three in the middle equation. If we add three to both sides, we get X is equal to positive three. And on the right hand side, if we subtract two to both sides, we're going to get X is equal to negative two. So in total, we have three zeros of X and we're gonna go ahead and plot these zeros on a number line. This number line is going to represent the domain of the polynomial function and it's gonna reach from negative affinity to positive affinity. Now, what we're gonna do is put the zeros on this number line. So we're going to have negative three, we have negative two and we have positive three. Now, what we wanna do is we want to go ahead and pick a testing point from each of the regions between the zeros in order to find the regions of increasing and decreasing. Now, there is a trick that we can do in order to find the increasing decrease in regions of the end points. If we take a look at the first term in the polynomial function, notice that we have an exponent of an odd number. So that tells us that this function is going to be odd. Furthermore, the first term is positive. So that means that the end behaviors of the graph as X approaches positive infinity, the graph is going to increase to positive infinity. And as X approaches negative infinity, the graph is going to decrease to negative infinity. So just based off of the first term, we already know the behaviors of the end behaviors of the graph. So now what we need to do is we need to take a testing point between negative three and negative two and between negative two and three and see what number C it produces in order to determine the increasing or decreasing regions. So let's go ahead and let's take the testing point of X is equal to negative one. If X is equal to negative one, we can go ahead and plot that either in the original polynomial function or the factored version of the polynomial function. Let's go ahead and plug this into the factor version of the polynomial function. So when X is equal to negative one, we get negative one plus three, which is going to give us two times negative one minus three, which is negative four times negative one plus two, which is going to be positive one, two times, negative four times one is going to give us negative eight. And notice that when X is equal to negative one, we get a negative number. So that means that the region between negative three and negative two is going to decrease. Now let's take a testing point between negative two and three. A testing point in this region can be when X is equal to zero. When X is equal to zero. And we plug it into the factored form of the expression. This is going to give us zero plus three, which is three times zero minus three, which is negative three times zero plus two, which is just two and three times negative three times two is going to give us negative 18, which is also a negative number. So that tells us that the region of this graph is going to decrease. So now that we have the regions of increasing and decreasing, let's go ahead and find the Y intercept because we're going to need that for the graph. So whenever we're looking for a Y intercept, we're looking for the region of the graph where it intersects on the Y axis. And this is when X is going to equal to zero. But we already know the value of when X is equal to zero, it's going to be negative 18. So if we plug in zero into the function, we're going to get the value of negative 18, which means that we have a Y intercept at zero comma negative 18. So now that we have this information, let's go ahead and make a rough sketch of the graph of the function. So we're going to first start by plotting our zeros, we have a zero and negative three one at negative two and we have a zero at positive three, we also have the Y intercept at negative 18. Now we know the first N be, we know the end behaviors of the graph, we know that as X approaches negative infinity, this graph is going to decrease to negative infinity. And we know that as the graph approaches positive infinity, the graph is going to increase to positive infinity. So the region between negative three and negative two is going to decrease. And we can represent that as a small arc decreasing between negative three and negative two. And we also know that the graph is going to decrease between negative two and three. So we can go ahead and draw an upside down hill that also intercepts at the Y axis. And this is going to be the rough sketch of the graph of the polynomial function. So now what we need to do is just pick the option that accurately represents this information. And that option is going to be option B just to verify we have the zeros at negative three, negative two and positive three, we have the white intercept at negative 18 and the shape of the graph roughly matches the one that we drew. So with that being said, the answer to this problem is going to be B. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=x^3+5x^2-x-5

Key Concepts

Polynomial Functions

Factoring Polynomials

Graphing Techniques

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