In Exercises 25–32, find the zeros for each polynomial function a...

Question

In Exercises 25–32, find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=−3(x+1/2)(x−4)^3

Accepted Answer

Welcome back. I'm so glad you're here. We are given the function F of X equals negative seven times parentheses. X plus 3/5 and parentheses? New parentheses? X minus nine And parentheses to the third power. And we are asked to find several things here. First we need to find the zeros of the polynomial and then we need to state the zeros multiplicity ease. And then finally we need to describe how this graph behaves with respect to the X axis. So let's start off with the zero's first step in finding the zeros as we need to factor our function. And yeah, it's already factored for us. So now we can take each of these factors X plus 3/ X minus nine and we can set them equal to zero to find out what our zeros are. What values for X would give us zero. And one thing to note is that there are three x minus nines that we can set equal to zero. We can just solve for it one time. But it is important to note that there are three of them. So solving for the first one, X plus 3/5 equals zero. Let's subtract 3/5 from both sides and we get x equals a negative 3/5. There's one of our zeros for X minus nine equals zero. Will add nine to both sides and we get X equals nine. And we get that three times we get X equals nine. X equals nine. X equals nine, awesome. All right. We found our zeros. Now we are going to find their multiplicity ease. We recall from previous lessons that multiplicity is. How many zeros of the same kind There are. So how many negative 3/5 are there? There's just one. So the multiplicity of the zero negative 3/5 equals one. Now, how many nines are there? Well there are three of them. So the multiplicity for the zero X equals nine is three. Alright. We found their multiplicity is. Now we need to describe what's going on with the X axis. We recall from previous lessons that when the multiplicity is odd then the graph function line is just going to pass right through the X axis at that X value. If the multiplicity is even then the grafts function line is going to boop it. It's going to touch it and then it's going to turn around at that X value on the X axis. So let's look at each of our multiplicity is we have a multiplicity of one. That is odd. We have a multiplicity of three. That is also odd. So at both negative 3/5 and nine. The function line is going to pass right through the X axis at those two values. Let's look at our answer choices. This matches with answer choice. A well done. We'll catch you on the next one

In Exercises 25–32, find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=−3(x+1/2)(x−4)^3

Key Concepts

Zeros of a Polynomial

Multiplicity of Zeros

Graph Behavior at Zeros

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