finding zeros given the graph of a polynomial function chapter 5.6
TRANSCRIPT
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
• 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
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
• 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
• 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
• 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
• 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
• 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
• 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
• 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