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-1)(x+2)^2

Accepted Answer

Hello. Today we're gonna be graphing the following polynomial function. So what we are given is F of X is equal to five X minus two times the quantity X plus one squared. Now, in order to graph this polynomial function, we're going to need to find the regions of increasing and decreasing for the function. But in order to do that, we're going to need to first get the zeros of X. So in order to get the zeros of X, we're going to take our function, which in this case is five X minus two times the quantity X plus one squared. And we're going to set it equal to zero and solve for the values of X. Now notice that our function is already in a fully factored form. So what we can go ahead and do is we can use our zero property to set each factor of X equal to zero. So doing this is going to give us two equations. We're going to get five X minus two is equal to zero and we're going to get X plus one squared equal to zero. On the left hand side, we're going to first add two to both sides. In order to sell for X, that is going to give us five, X is equal to two, then we're going to divide both sides of the equation by five. And that is going to give us X is equal to 2/5, which is going to be the first zero of X on the right hand side. In order to solve for X, we're first going to need to eliminate the exponent of two. And in order to do that, we can go ahead and take the square root of both sides of the equation that is going to leave us with X plus one is equal to zero. And finally, if we subtract one to both sides of the equation, what we are left with is X is equal to negative one, which is going to be the second zero for the function. So now that we have the zeros for the function, we're going to go ahead and plot the zeros on a number line, this number line is going to represent the domain going from negative infinity to positive infinity. And again, we're plotting the zeros. So we have negative one and 2/5 on the number line. So the next step is to go ahead and take a testing point around and between the zeros of X plug it into the function and see what sign the function produces. That is going to tell us whether we have a region of increasing or decreasing. So what's the point that I can pick from the region that comes before negative one? Well, I can go ahead and pick the testing point X is equal to negative two. Since that comes before negative one on the number line. If I plug that into the original function, what I get is five times negative two which is negative 10 minus two times the quantity negative two plus one, which is negative one squared, negative 10 minus two is going to give me negative 12 and negative one squared is going to give me positive one. Finally negative 12 times one is going to give me negative 12. So notice that when we plug in the value of X is equal to negative two into the original function, we get an output of negative 12. Since the value is negative, that tells us that the region is going to be decreasing. And now we're gonna repeat this process for the region between negative one and 2/5. Well, a testing point that we can pick from this region is going to be when X is equal to zero. If we plug in zero into the function, we get five times zero, which is zero minus two, which is negative two times zero plus one squared, which is going to give us one squared. Now one squared is just going to give us positive one and positive one times negative two is going to give us negative two. So that tells us that the region between negative one and 2/5 is going to be decreasing. Now, we're gonna pick a testing point that comes after 2/5. Well, that testing point can be when X is equal to one, when X is equal to one, we get five times one which is five minus two times the quantity one plus one, which is two squared. Now five minus two is going to give us three and two squared is going to give us four, finally, four times three is going to give us 12. And since this produces a positive number that tells us that the region that comes after 2/5 is going to be increasing. Now we need one more piece of information for the graph and that is going to be the Y intercept. Now, in order to find the Y intercept, we need to go ahead and plug in the value of zero into the function. Since the Y intercept lies in the region when X is equal to zero. But notice that we've already used X is equal to zero as a testing point. When we plug in zero to the function, we get the value of negative two. So that tells us that the Y intercept is going to be zero, comma negative two. So now that we have the regions of increasing and decreasing and we have the white intercept, we can go ahead and use this information to make a rough sketch of the graph. So the first thing we're gonna do is we're going to plot our zeros, we have the zero of negative one and we have the zero of 2/5, we also have the Y intercept at negative two. So for the regions, we know that the region as X approaches negative infinity is going to decrease to negative infinity. So that's going to give us an end behavior on the left hand side that decreases to negative infinity. We also know that the region between negative one and 2/5 is going to decrease and also intercept the Y axis and negative two. So that portion of the graph is gonna roughly look something like this. And finally, the region as X approaches positive infinity, which is after 2/5 is going to increase. So that tells us that the graph is going to increase the positive infinity as it approaches infinity. So this is going to be the rough shape of the graph for the polynomial function. And now we just need to pick the option that accurately represents this information. While the option up above that accurately represents this is going to be D. And just to verify we have the zeros at negative one and 2/5, we have a Y intercept at negative two. And the shape of the graph roughly matches the one that we drew below So with that being said, the answer to this problem is going to be d 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.

My Courses

Chemistry

Biology

Math

Physics

Business

Social Sciences

Programming

Product & Marketing

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=(3x-1)(x+2)^2