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)=2(x−5)(x+4)^2

Accepted Answer

Welcome back. I'm so glad you're here. We are given a function F of x equals five times parentheses X minus three times parentheses. X plus six squared. We're asked to find several things associated with this function. First. We are to find the zeros and then we are to find the zeros. Multiplicity ease. And then were asked to describe what's going on with this graph at the X axis. So let's start off with the zero's first step in finding zeros of a polynomial function is to factor our function and it's already been factored for us. Great. Now let's set each of those factors that includes an X equal to zero. So x minus three equals zero and X plus six equals zero. And we note that there are two X plus six factors. Now let's solve for X and find out what values for X will give us zero in our polynomial function. First we have X equals three. A three would give us a zero. There's one of them. And next we have X equals negative six. Negative six. Would give us a zero. All right. We found our two and we note that there are two for X equals negative six. Okay, we found our zeros. Next their multiplicity ease. We recall from previous lessons that multiplicity is just how many of each? Zero are there? So how many threes are there? Well, there is one, X equals three. So X equals three is a zero with a multiplicity of one. How about negative six. How many negative six is are there? Well, there are two negative six is so X equals negative six is a multiplicity is zero with a multiplicity of two. Great. We found our multiplicity is Now let's describe how the graph is behaving at the X axis. We recall from previous lessons that when the multiplicity of a zero is odd, then the graphs line will pass right through the X axis at that zero. When the multiplicity of a zero is even. However, then the graph will just boop the X axis at that zero. It will touch the X axis at that zero and then it will turn around. So let's take a look at each of our zeros and their multiplicity is for X equals three. That is a zero with the multiplicity of one and one is odd. So at X equals three. The function line is going to pass right through the X axis. Now for X equals negative six. That is a zero with the multiplicity of 22 is even so at X equals negative six. The function line is going to touch and then turn around at the X axis. We look at our answer choices and this matches with answer choice. C. Well done. We'll catch 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)=2(x−5)(x+4)^2

Key Concepts

Zeros of a Polynomial

Multiplicity of Zeros

Graph Behavior at Zeros

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