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)=(4x+3)(x+2)^2

Accepted Answer

Hello. Today we're gonna be graphing the following polynomial function. So what we are given is two X plus five times the quantity X plus one squared. Now, in order to graph this polynomial function, we're going to need to identify the regions of increasing and decreasing. But in order to do that, we're going to need to first find the zeros of X. So what we're going to do is we're going to take our polynomial function which again is two X plus five times X plus one squared. Then what we're going to do is set this equation equal to zero, then solve for X. Now notice that the function is already in a factored form. So since the function is fully factored, we can take each factor of X and set it equal to zero using our zero property. So doing this is going to give us two X plus five is equal to zero and the quantity X plus one squared is equal to zero. Now, what we need to do is sol for X in each of the equations. So on the left hand side, in order to solve for X, we're going to subtract five to both sides of the quantity. And that is going to give us two, X is equal to negative five. Then finally, we're going to divide both sides of the equation by two. And that is going to give us X is equal to negative 5/2, which is going to be the first zero of X that we're going to need. Then on the right hand side, we're going to need to solve for X. But in order to do that, we need to eliminate the exponent of two. In order to eliminate the exponent, we can take the square root of both sides of the equation. Doing so is going to leave us with X plus one is equal to zero. And 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 X. Now that we have the zeros for X, we're gonna go ahead and plot the zeros on the number line. This number line represents the domain of all the X values of the graph going from negative infinity to positive infinity. Then what we're gonna do is we're going to put the zeros on the graph. So we have negative 5/2, which is also approximately equal to negative 2. and we have the point negative one. So now what we're going to do is we're going to take a testing point from each region around the zeros. And we're going to plug it into the original function to see what kind of sign that we get. And depending on that sign that is going to tell us whether we have increasing or decreasing region. So what's a testing point that we can take? That is less than negative 2.5. Well, we can go ahead and pick X S equal to negative three for that region. If we take negative three and plug it into the equation, we get two times negative three, which is negative six plus five times the quantity negative three plus one, which is negative two squared. Now negative six plus five is going to give us negative one and negative two squared is going to give us positive four and finally, four times negative one is going to give us negative four. Now notice that when we take X is equal to negative three and plug it into the equation that is going to give us a negative number. So since we have a negative number that tells us that the region before negative 5/2 is going to be decreasing. Now let's pick a testing point between negative 5/2 and negative one. While that value can be when X is equal to negative two. If we plug this into the equation, we get two times negative two, which is negative four plus five times negative two plus one. Which is negative one squared, negative four plus five is going to give us positive one and negative one squared is also gonna give us positive one. Now what we are left with is one times one, which is just going to give us one. So since we got a positive number by taking a testing point between this region that tells us that the region between negative 5/2 and negative one is going to be increasing. Now, we just need to take a testing point that comes after negative one. And that can be when X is equal to zero, when X is equal to zero, what we are left with is two times zero, which is zero plus five, which is just gonna be five times zero plus one, which is one squared. Now we know that one squared is just going to give us one and five times one is going to give us five. So since we have a positive number that tells us that the region that comes after negative one is going to be increasing, now we do need one more thing for the graph and that is going to be the Y intercept. When we are looking for the Y intercept, we are looking for the Y intercept point of the function. And that is going to be the region when X is equal to zero. But notice that we've already plugged in zero into the original function that gave us a value of five. So that means that the Y intercept is going to be zero comma five. So now that we have the Y intercept and the regions of increasing and decreasing, 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 negative 5/2 and we have negative one. And we also know that we have a Y intercept at zero comma five. So now we're going to go ahead and plot the end behaviors. We know that as X approaches negative infinity, the graph is going to be decreasing to negative infinity. We also know that as X approaches or X approaches positive infinity, the graph is going to be increasing to positive infinity. But we know that the graph is going to decrease between negative 5/2 and negative one. And that's gonna look like a negative slope between negative 5/2 and negative one. So this is going to be the rough sketch of the function graph. And now we just need to go ahead and pick the option that accurately represents this information. Well, that option in this case is going to be a and just to make sure we have the zeros at negative 5/2 and at negative one, we also know that the Y intercept is at zero comma five 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 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)=(4x+3)(x+2)^2

Key Concepts

Polynomial Functions

Factoring Polynomials

Graphing Polynomial Functions

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