In Exercises 30–31, find the zeros for each polynomial function and give the multiplicity of eac

In Exercises 30–31, find the zeros for each polynomial...

Key Concepts

Graph Behavior at Zeros

The behavior of the graph of a polynomial at its zeros depends on the multiplicity of each zero. If a zero has an odd multiplicity, the graph crosses the x-axis at that point, whereas if the multiplicity is even, the graph touches the x-axis and turns around, without crossing it.

Multiplicity

Multiplicity refers to the number of times a particular zero occurs as a solution of the polynomial equation. It is determined by the exponent on the factor corresponding to that zero and affects the behavior of the graph near the zero.

Polynomial Function

A polynomial function is an expression that consists of variables raised to whole-number exponents and multiplied by coefficients, summed together. Its structure allows it to be factored into products of simpler polynomials, which is useful for finding its zeros.

Zeros of a Polynomial

The zeros (or roots) of a polynomial are the values of the variable that make the polynomial equal to zero. Finding the zeros involves setting each factor of the fully factored polynomial equal to zero and solving for the variable.

Transcript

00:01 Okoky, we want to find all the zero values for this function, the multiplicity of those zeros, and whether or not it just touches the x -axis, or it crosses.

00:11 Now, finding the zeros, very easy.

00:14 Since we've got like this, take it we've got is this, this, multiply by this, multiplied by this, multiplied by this.

00:19 And the whole thing will be zero when one of these functions equals zero, because if this equals zero, well, anything times zero is zero.

00:27 So to get this to zero, all we need to do is make x equal to 1, because 1 minus 1, well, that's 0.

00:34 So this is going to be 1.

00:35 This one, to make it 0, 2, minus 2 is obviously going to be 0.

00:38 So this one's going to be minus 2.

00:40 And i'm just going to bring this power down here and put it like this, okay? and this one, i hope you can see, it's going to be minus 5.

00:48 And this is 3.

00:50 Now, these are our 0 values here.

00:55 1 minus 2, 1, minus 2.

00:58 And minus five.

01:00 And their multiplicities are simply the powers of top.

01:03 Sort of how many times they occur.

01:06 So this one will be one has a multiplicity of one, minus two is a multiplicity of two and minus five as a multiplicity of three.

01:15 One, two, and three.

01:16 Now the final thing to determine is whether or not they cross the x -axis or they just touch the x -axis.

01:24 Now, i hope you can see that if, or also, it could be that they touch the x -axis like this.

01:30 Now, if you can see, that our output, both here, they both stay negative.

01:36 Here, they both stay positive.

01:38 And equally on here, they change, and if it was coming down with like this, crossing the x -axis like this, again, it goes from positive to negative, here from negative to positive, here they both stay negative, and here they both stay positive.

01:49 So if we input a value just above our 1 and just below our 1, well, we should be able to see whether or not it touches or thing, similarly with our minus 2 and our minus 2, minus 5.

01:59 So if into our function we put in, let's say, let's put in zero.

02:06 Well, that's just before 1, isn't it? if we put in 0 into this, well, this is all going to come out, this is going to come out at minus 2 lot times minus 1 times, well, 0 plus 2 is 2, so times 2 square times 5 cubed.

02:28 And that will actually come out at a thousand.

02:31 So this is our function of x.

02:35 The function at zero is equal to a thousand.

02:42 Now, if we were going to put in a number just above our one now, let's say two...

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