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)=3x^4-7x^3-6x^2+12x+8

Accepted Answer

Hello. Today we're gonna be graphing the following polynomial function. So the function given to us is four X cubed plus three X squared minus nine X plus two. Now, the first thing we're gonna want to do in order to graph this function is to figure out the intervals of increasing and decreasing. But in order to do that, we're going to need to get the zeros of X. So what we're going to do is set the equation equal to zero. Then we want to solve for the values of X. Currently, the function is not factor by factoring by grouping. So what we're going to do is we're going to use the method of synthetic division to simplify the given function. So we're going to list out the factors of the last term which we're going to call the P factors and those factors are just going to be one and two. Then we're going to list out the factors of the first term which you're going to call the Q factors and those factors are going to be 12 and four. Then through the P Q method which is P over Q, we're going to list all the possible zeros for the synthetic division. And that's going to be the P factors, which is 1/1 and two over the Q factors, which is gonna be 12 and four through simplification. This is the total list of possible zeros is going to be 121 half and 1/4. So now what we need to do is we need to pick one of the possible zeros to synthetically divide the polynomial by. Well, let's go ahead and try positive one for the synthetic division. So we're going to write number one in the zero spot. And then we're gonna list out the coefficients of the polynomial which are going to be 43, negative nine and two. In order to do the synthetic division, we're going to bring down the first term, first term which is going to be four. Then we're gonna multiply this term by the possible zero that we chose, which in this case is one. So four times one is going to give us four. And that's what we're going to put under. The next term in the synthetic division four plus three is going to give us seven. Then we're going to repeat this pattern of multiplying the next term by the possible zero, we chose seven times one is going to give us seven. And so what we have is negative nine plus seven, which is going to give us negative two, the negative two times one is going to give us negative two, which is going to give us zero. So the remainder that we have is zero, which means that the synthetic division worked. So now what we're going to do is we're going to list out a new polynomial with one power less using the new terms that we came up with. And that new polynomial is going to be four X squared plus seven X minus two. Now keep in mind that we use the possible zero of one. So what we're seeing here is that X is equal to one, but we want to list out the factor of X, that's going to give us one. Well, if we subtract one to the left hand side of this equation, we get X minus one is equal to zero. And that means that we factored out the quantity X minus one in order to perform the synthetic division. Now, the quantity four X squared plus seven X minus two is in the form of a factor trinomial and that trinomial is going to factor to give us the quantities of four X minus one times X plus two. And keep in mind that we still have the factor of X minus one from the synthetic division. Now keep in mind that these quantities of X are still equal to zero. So using the zero property, we can go ahead and set each of the factors of X equal to zero. And sulfur X in order to get the zeros, so we have X minus one is equal to 04, X minus one is equal to zero. And X plus two is equal to zero on the very left. If we add one to both sides of the equation, we get X is equal to one which is going to be the first zero of X. For the graph. In the middle equation, we add one to both sides of the equation to give us four, X is equal to one. And in order to solve for X, we divide both sides of the equation by four. And that is going to give us X is equal to 1/4, which is another possible zero for X. Then on the right hand side, we subtract two to both sides of the equation. And that is going to give us X is equal to negative two, which is going to be the final zero for X. So now that we have the zeros for X, we can plot this on a number line in order to find the intervals of increasing and decreasing for the polynomial function. So if we draw a number line, this number line is gonna be going from negative infinity to positive infinity. And we're gonna list out the zeros that we just came up with. So we're gonna list negative two, we're going to list 1/4 and we're going to list one. Now what we need to do is we need to pick some testing points between each of those zeros that way, no matter what number we come up with, we can use that to determine whether the region is going to be increasing or decreasing. Now, here's a shortcut for getting the regions of increasing decreasing for the end points. Let's take a look at the first term of the original polynomial. The exponent is three, which indicates that this is going to be an odd function. And because the leading coefficient is positive, this is going to be an odd function with the positive coefficient. So the end points are going to be decreasing when F FX is approaching negative infinity and increasing when F of X is approaching positive infinity. So just knowing the first term in the polynomial, we can conclude that as X approaches negative infinity, the graph is going to decrease. And as X approaches positive infinity, the graph is going to increase. So now what we need to do is we just need to test the regions between negative two and 1/4 and 1/4 and one. Let's go ahead and start with this region here. What is a possible testing point that we can use that lies between negative two and 1/4? Well, we can go ahead and use X is equal to X is equal to zero and you can go ahead and plug this in into either the original polynomial or the factored form of the polynomial. If we plug this into the original polynomial, any term that has an X in it is going to turn to zero. So that means that we have four times zero plus three times zero minus nine times zero plus two. And that is going to produce the value of two. Now, we don't care about what the actual value of this point is going to be. But because two is a positive number that tells us that the region of this graph is going to be increasing. So we only care about the sign of the value that's produced when we plug in X is equal to zero into the function. Now, we need to pick a testing point between the region of 1/4 and one. While a possible testing point is going to be when X is equal to 0.5, we can go ahead and plug this into the factored form of the polynomial. If we plug in positive 0.5 into the factored form, we get 0.5 minus one, which is going to give us negative 0.5, then we have four times 0.5, which is going to give us two minus one, which is going to give us one, then we have 0.5 plus two, which is going to give us 2.5. Now negative 0.5 times one times 2.5 is going to give us the value of negative 1.25. But because this produced a negative number that tells us that the region between 1/4 and one is going to be decreasing. So now we have the regions of increasing and decreasing for the graph, there's one more thing we need to find. And that's going to be the Y intercept of the graph. In order to get the Y intercept, we need to plug in zero into the function. And the reason for that is because the Y intercept is going to lie on any point along the Y axis. And this is the region where X is equal to zero. So if we plug in zero into the original polynomial, we get F of zero is equal to four times zero, which is zero plus three times zero, which is zero minus nine times zero, which is zero and then plus two. So that means that we're going to get the value of two for the Y intercept. And that's going to give us the point of zero comma two for the Y intercept. So now that we have the regions of increasing decreasing and the white intercept, we can go ahead and use this information to create a rough sketch of the graph. We start by listing the zeros of X on the X axis and that's going to be a negative two at 1/4 and at one, and we know that we have a Y intercept. At zero comma two. So now we're gonna go ahead and follow along with the regions of increasing and decreasing. So as the graph approaches negative two from negative infinity, the graph is going to be decreasing. So we're going to have an end point, an end point that decreases to negative infinity. Then the next region of the graph going from negative two to positive 1/ is going to be increasing. So that's going to create a rough sketch of a positive looking slope. Then the region going from 1/4 to 1 is going to be decreasing. So the sketch is going to dip below the X axis, then reach to one. And finally, the region from one to positive infinity is going to be increasing, which means that we're going to have an end point that goes up to positive infinity. And this is going to be the rough sketch of the graph. So now that we have the rough sketch of the graph, we can use this to conclude the solution to this problem. And the graph that accurately represents all this data is going to be option A and just to verify we have the end points at negative two, 1/4 and at one, we have a Y intercept at two and the region or the s the sketch itself roughly matches the graph that we came up with below. So the solution to this problem is going to be A. 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)=3x^4-7x^3-6x^2+12x+8

Key Concepts

Polynomial Functions

Factoring Polynomials

Graphing Polynomial Functions

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