Finding Zeros Given the Graph of a Polynomial

FunctionChapter 5.6

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 function

• These are the points where the graph intersects the x-axis

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 equation

• The next slide shows the graph of and the position of the zeros

Review: Zeros of Quadratic Functions

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 possible• Use the quadratic formula• Complete the square

• It so happens that this expression is factorable and can be written as

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 equations

• The solutions are therefore

• Note that these are the positions on the x-axis where the graph intersects the axis

Review: Zeros of Quadratic Functions

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 polynomial

• This means that we can write as

• The factor is another polynomial

• Our goal in this lesson is to find the missing zeros

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 zero

• We 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

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 zeros

• To find zeros you will be given one (sometimes two) zeros

• These 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

Finding Zeros From a Graph

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 x

• You should get

Finding Zeros From a Graph

Finding Zeros From a Graph

• The next example is a 4th degree polynomial

• Note that the degree tells us the maximum number of zeros that the function can have

• If 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 solution

• The examples shown and the problems you will work for practice will give two zeros rather than just one

Finding Zeros From a Graph

• Find the zeros for the 4th degree polynomial function

• The graph of the function, along with the known zeros, are shown on the next slide

Finding Zeros From a Graph

Finding Zeros From a Graph

• Since the zeros are , then the function can be written as

• In order to find the polynomial , you may• Multiply to get a quadratic expression, then use long division• Use synthetic division twice: once for and once for • In most cases, synthetic division is the easier choice

• Use long division to divide

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 same

• You should find that we again obtain

• Use the square root property to solve

• The missing zeros are

Finding Zeros From a Graph

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 2

• When this occurs, you should either square the known factor and use long division, or use synthetic division two times using k

• Here is an example using a cubic polynomial:

Finding Zeros From a Graph

Finding Zeros From a Graph

• The function factors as

• You can use long division by first squaring , but using synthetic division twice is easier

• The missing factor is

• Since this is the only other factor, we can easily solve the equation

• The solution is

Finding Zeros From a Graph

Finding Zeros From a Graph

• This last example is a 4th degree polynomial with a known zero of multiplicity two

• The function is

• Note again (on the next slide) that the zero occurs at a turning point of the graph

• In fact, both zeros occur at turn points, so the other zero is also of multiplicity 2

Finding Zeros From a Graph

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 expression

• We get

• Set this equal to zero and solve

• You should get

Finding Zeros From a Graph

Exercise 5.6a

• Handout