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+5)(x+2)^2

Accepted Answer

Welcome back. I'm so glad you're here. We are given the function F of X equals five times parentheses X plus six. And parentheses new parentheses X plus eight end parentheses squared. And we're asked to find several things here. We need to find the zeros of this polynomial function and then find the zeros multiplicity ease. And then we are asked to describe how the graph behaves with respect to the X axis. Alright, let's start with the zero's first step to finding zeros of a polynomial function is to factor our function and it's already factored for us. So we're going to set each of these factors X plus six and X plus eight equal to zero. And we note that there are two X plus eight factors. Now we can solve for X to find out what values of X would make our function equal to zero. So for the first X plus six that's going to be an X equals negative six, gives us +10. And for the second one here we get an X equals negative eight. And we note that there are two negative eights. All right, we found our zeros. Now their multiplicity ease. We recall from previous lessons that multiplicity, That's just how many of that particular zero are there? And for X equals negative six. There's only one negative six. So negative six has a multiplicity of one. For negative eight. There are two X equals negative eights. So negative eight has a multiplicity of two. Great. We have found their multiplicity is. Now what does that have to do with what's going on in the X axis? We recall from previous lessons that when a multiplicity is odd, that means that the graph line is going to pass right through the X axis at that zero value. If the multiplicity is even then the graph line is just going to boop the X axis at that zero's value or it's going to touch that value and then turn around. So for X equals negative six. That's a multiplicity of one. Which is odd. So at X equals negative six. The graph line is going to pass right through, I don't know if it's coming from above or below but it's going to pass right through the X axis at negative six. And for X equals negative eight. It has a multiplicity of two that is even two is even so at X equals negative eight. The graph line is just going to boop at negative eight on the X axis. And then turn around. It's not going to pass through. Might come from above or below but it will touch it. And then turn around. We look at our answer choices where X equals negative six has a multiplicity of one. X equals negative eight has a multiplicity of two. And it's going to pass through at negative six. And touch and turn around at negative eight and that matches with answer choice. D 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+5)(x+2)^2

Key Concepts

Zeros of a Polynomial

Multiplicity of Zeros

Graph Behavior at Zeros

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