Finding Zeros Given the Graph of a Polynomial Function - PowerPoint Presentation

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Finding Zeros Given the Graph of a Polynomial Function

Chapter 5.6Slide2

Review: Zeros of Quadratic Functions

In the previous chapter, you learned several methods for solving quadratic equations

If, rather than a quadratic equation

, we think about the function , then setting this equation equal to zero is the same as setting On the graph of a function, the value(s) of where are called the zeros (or roots or x-intercepts) of the functionThese are the points where the graph intersects the x-axis

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Review: Zeros of Quadratic Functions

Suppose you are required to find the zeros of the quadratic function

Since the zeros are the points on the graph where

, then you would find the zeros by solving the equationThe next slide shows the graph of and the position of the zeros Slide4

Review: Zeros of Quadratic FunctionsSlide5

Review: Zeros of Quadratic Functions

In solving the equation

we are looking for the

x-coordinates of the two points (we already know the y-coordinates; what are they?)You have learned several methods for solving the above equation:Factor the expression, if possibleUse the quadratic formulaComplete the squareIt so happens that this expression is factorable and can be written as Slide6

Review: Zeros of Quadratic Functions

To solve the equation

, we can use the Zero Product Property that says that, if two numbers are multiplied and the result is zero, then one or the other of the numbers must be zero

So, we split the equation into two separate equationsThe solutions are therefore Note that these are the positions on the x-axis where the graph intersects the axis Slide7

Review: Zeros of Quadratic FunctionsSlide8

Finding Zeros From a Graph

The Factor Theorem tells us that, for a polynomial function

, if we know of some number

such that , then is a factor of the polynomialThis means that we can write asThe factor is another polynomialOur goal in this lesson is to find the missing zeros

 Slide9

Finding Zeros From a Graph

The zeros of a function occur at those values of

where

Since , then as we did with the quadratic function example, we set the right side equal to zeroWe can use the Zero Product Property and create two separate equations

We already know that one zero was

, but we cannot solve the other until we know what

is

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Finding Zeros From a Graph

We will be able to find

by synthetic division

Notice, however, that there is a kind of “cheat” to this method because we must already know one of the zerosTo find zeros you will be given one (sometimes two) zerosThese are the k values that we can then use to find the polynomial Let’s see how this works: the next slide shows the graph of the 3rd degree polynomial function Slide11

Finding Zeros From a GraphSlide12

Finding Zeros From a Graph

Use synthetic division to find the missing factor

The missing factor is the quadratic polynomial

To find the zeros, set the expression equal to zero and solve for xYou should get  Slide13

Finding Zeros From a GraphSlide14

Finding Zeros From a Graph

The next example is a 4

th

degree polynomialNote that the degree tells us the maximum number of zeros that the function can haveIf you are given only one of the zeros for a fourth degree polynomial, then you would have to solve a cubic equation, and we have no easy way to solve this without already knowing a solutionThe examples shown and the problems you will work for practice will give two zeros rather than just oneSlide15

Finding Zeros From a Graph

Find the zeros for the 4

th

degree polynomial functionThe graph of the function, along with the known zeros, are shown on the next slide Slide16

Finding Zeros From a GraphSlide17

Finding Zeros From a Graph

Since the zeros are

, then the function can be written as

In order to find the polynomial , you mayMultiply to get a quadratic expression, then use long divisionUse synthetic division twice: once for and once for

In most cases, synthetic division is the easier choice

Use long division to divide

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Finding Zeros From a Graph

Using long division we get

To compare methods, use long division twice

You may start with either or , the result will be the sameYou should find that we again obtain Use the square root property to solve The missing zeros are  Slide19

Finding Zeros From a GraphSlide20

Finding Zeros From a Graph

When the zero of a function is found at a point where the graph of a polynomial function turns, then the function may factor as

In a case like this, we must count

k as occurring twice; we say that k is a zero of the function of multiplicity 2When this occurs, you should either square the known factor and use long division, or use synthetic division two times using kHere is an example using a cubic polynomial: Slide21

Finding Zeros From a GraphSlide22

Finding Zeros From a Graph

The function factors as

You can use long division by first squaring

, but using synthetic division twice is easierThe missing factor is Since this is the only other factor, we can easily solve the equationThe solution is

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Finding Zeros From a GraphSlide24

Finding Zeros From a Graph

This last example is a 4

th

degree polynomial with a known zero of multiplicity twoThe function isNote again (on the next slide) that the zero occurs at a turning point of the graphIn fact, both zeros occur at turn points, so the other zero is also of multiplicity 2 Slide25

Finding Zeros From a GraphSlide26

Finding Zeros From a Graph

Use synthetic division twice with

to find the missing factor

The missing factor is This quadratic polynomial is factorable; use the usual method for factoring a quadratic expressionWe get Set this equal to zero and solveYou should get

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Finding Zeros From a GraphSlide28

Exercise 5.6a

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