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

Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4.

$ƒ (x) = - 3 x^{4} - 5 x^{3} + 2 x^{2}$

Accepted Answer

Hello. Today we're gonna be graphing the following polynomial function. So the polynomial function given to us is negative X cubed plus two X squared plus 15 X. Now, in order to start graphing this function, we need to first find the regions of increasing and decreasing of the function. But in order to do that, we first need to find the zeros of X. So what we're going to do is we're going to take our function and set it equal to zero. Then we want to go ahead and we want to factor the following function. So notice that each term in the given function has at least one factor of X. So that means that we can go ahead and factor out an X from the given polynomial function. But when we factor out an X, we do want this leading coefficient to be positive. So what we're going to do is we're going to factor out a negative X from the given given function. And that is going to leave us with X squared minus two, X minus 15 is equal to zero and X squared minus two X minus 15 is in the form of a factor trinomial through the trial and error method. This is going to factor to give us X plus three times X minus five. And keep in mind that we still have this factor of negative X in front of the two groups. Now using our zero property, we can set each factor of X equal to zero and solve for the corresponding values. So we're going to get negative X is equal to zero, X plus three is equal to zero and X minus five is equal to zero. Then we just have to solve for X. And each of the following equations. In the first equation, we just divide both sides of the equation by negative one. And that is going to give us X is equal to zero, which is the first zero for X. In the second equation, we subtract three from both sides and that is going to give us X is equal to negative three. And in the third equation, we just add five to both sides of the equation. And that is going to give us X as equal to positive five. So in total, we have three zeros for X. And now what we want to do is we're going to plot these zeros on a number line. This number line is gonna be going from negative infinity to positive infinity. And the zeros that we have for X are at negative 30 and positive five. Now what we want to go ahead and do is take a testing point between each of the regions and pay attention to the sign of the given output because the sign is going to tell us whether the regions are going to be increasing or decreasing. Now, there is a trick that we can use to get the, the end points of the graph. And that is to take a look at the leading term of the current track of the current polynomial function. So if we look at the exponent, the exponent is three, which is an odd number. So that tells us that the function is going to be an odd function. And that means that the end points are going to be facing in opposite directions. So normally a positive odd function has end behaviors of increasing and decreasing. However, since the leading coefficient is negative, that means that the end behaviors are going to be increasing when F FX approaches negative infinity and decreasing when F FX approaches positive infinity. So now that we know the end behaviors of the graph, we just need to find the behaviors between the region of negative three and zero and zero and five. So what's a testing point that we can use to test the region between negative three and zero? While the testing point in this case can be when X is equal to negative one, when X is equal to negative one, we're gonna go ahead and plot that or plug that into the original equation. And that is going to give us negative negative one cubed plus two times negative one squared plus times negative one negative one cubed is going to give us negative one and negative negative one is going to give us positive one, negative one squared is going to give us positive one and two times one is going to give us two and 15 times negative one is going to give us negative 15. So summing these three values together is going to give us an answer of negative 12. Now, since this testing point produced a negative number that tells us that the region between negative three and zero is going to be decreasing. And so now we want to go ahead and test the region between zero and five. And that can be when X is equal to one. Now when X is equal to one plugging that value into the original function is going to give us negative one cubed plus two times one squared plus 15 times one, one cubed is just going to give us one and negative one times one is going to give us negative one, one squared is going to give us one and two times one is going to give us two and finally 15 times one is just going to give us 15, negative one plus two plus 15 is going to give us the value of 16. And since that is a positive number that tells us that the region between zero and five is going to be increasing. So now we have the regions of increasing and decreasing for the graph. But we need to find one more thing and that is going to be the Y intercept. Now the Y intercept lies on the Y axis and that is going to be the region when X is equal to zero. So if we were to plug in zero for every X in the original equation, notice that all the terms are just gonna go to zero. So that means that the Y intercept is going to be located at the origin zero comma zero. Now that we have all the information we need, we can go ahead and make a rough sketch of the current graph. So we're going to start by plotting the zeros for X and that is going to be a negative three at zero, which is also the Y intercept. And at positive five, we know that this graph is going to increase as it goes to negative infinity. And we know that it's going to decrease when it goes to positive infinity, going from negative 3 to 0 is going to decrease. So we're going to draw a negative slope between negative three and zero. And we know the graph is going to increase from 0 to 5. So we're going to draw a positive slope between zero and five and this is going to be the rough sketch of the graph for the given function. So now what we need to do is we just need to go ahead and select the option that accurately represents this information. And that option is going to be C to verify we have the zeros at negative three at zero, which is also the Y intercept and at positive five and the shape of this graph roughly matches the one that we came up with. And that verifies that the answer to this problem is going to be C So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see all in the next video.

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

Key Concepts

Polynomial Functions

Factoring Polynomials

Graphing Polynomial Functions

Watch next