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Find Rational Zeros, II Objectives: 1. To use a graph to help find the zeros of a polynomial function 2. To use the Location Principle to help find the zeros of a polynomial function Warm-Up The points (2, −3) and (3, 5) are on the graph of a polynomial function f (x). What conclusion can you draw about the graph of f (x)? There must be a zero between 2 and 3 since the graph must cross the x-axis to connect these points. The Rational Zero Theorem If f(x) = anxn + … + a1x + a0 has integer coefficients, then every rational zero of f(x) has the following form: factor of constant term a0 p q factor of leading coefficient an Important note: These factors can be either positive or negative. The Rational Zero Theorem Here’s another way to think about The Rational Zero Theorem. Consider the function below: f ( x) 6 x3 x 2 47 x 30 In factored form: f ( x) (2 x 5)( x 3)(3 x 2) And here are the zeros: 5 2 x , 3, and 2 3 The Rational Zero Theorem Now work backwards from the factors: What would be the constant term and the leading coefficient? f ( x) 6 x3 x 2 47 x 30 f ( x) (2 x 5)( x 3)(3 x 2) (5)(3)(2) 30 (2 x)( x)(3x) 6 x3 Factors of 6 5 2 x , 3, and 2 3 Factors of −30 The Rational Zero Theorem Remember that The Rational Zero Theorem just helps you to generate a list of the possible rational zeros of a polynomial function (with integer coefficients). To verify each zero, you have to use synthetic division. Exercise 1 List the possible rational zeros of f(x) using the Rational Zero Theorem. f ( x) x 12 x 35x 24 3 2 Exercise 2 Find all real zeros of the function. f ( x) x 12 x 35x 24 3 2 Exercise 3 List the possible rational zeros of f(x) using the Rational Zero Theorem. f ( x) 2 x 5x 11x 14 3 2 Exercise 3 List the possible rational zeros of f(x) using the Rational Zero Theorem. f ( x) 2 x 5x 11x 14 3 2 Notice that when the leading coefficient is not 1, the number of possible rational zeros increases in an unsavory way. Surely there has to be a way to narrow the possibilities… Exercise 4 According to The Rational Zero Theorem, what is the maximum number of possible rational zeros for the polynomial function below? f ( x) x3 bx 2 cx d Exercise 5 Assume a has k factors and d has r factors. According to The Rational Zero Theorem, what is the maximum number of possible rational zeros for the polynomial function below? f ( x) ax3 bx 2 cx d Objective 1 You will be able to use a graph to help find the zeros of a polynomial function Facilitation, 1 (Cheating) The first way to facilitate the search for rational zeros involves looking at the graph of the polynomial. Facilitation, 1 (Cheating) Since zeros are the same thing as a graph’s xintercepts, finding the approximate location of an x-intercept can give a reasonable place to start verifying factors from our long, tedious list of possibilities. Exercise 6 Find all real zeros of the function. f ( x) 2 x3 5x 2 11x 14 Facilitation, 1 (Cheating) Using the graph of a polynomial function to help finding its zeros seems a bit like cheating. This is because we want to find the zeros (or x-intercepts), and we do this by looking a graph’s xintercepts. If we already have a graph, then why are we trying to find the graph’s x-intercepts algebraically? (A fine example of circular logic) Exercise 7 Find all real zeros of the function. f ( x) 48x3 4 x 2 20 x 3 Exercise 8 Find all real zeros of the function. f ( x) 2 x 4 5x3 18x 2 19 x 42 Objective 2 You will be able to use the Location Principle to help find the zeros of a polynomial function Location Principle Another way to narrow 𝑥 𝑓(𝑥) down our rational zero 0 −6 choices is by using the 1 −12 Location Principle. 2 28 Consider the table of 3 150 values. What can you 4 390 conclude about the 5 784 location of one of the There must be a zero here zeros of f(x)? Location Principle Location Principle If 𝑓 is a polynomial function and 𝑎 and 𝑏 are two numbers such that 𝑓 𝑎 < 0 and 𝑓 𝑏 > 0, then 𝑓 has at least one zero between 𝑎 and 𝑏. For example, if 𝑓(2) = −3 and 𝑓(3) = 5, then there must be a zero between 𝑥 = 2 and 𝑥 = 3. This is because a continuous curve would cross the 𝑥-axis when connecting these points. Location Principle Location Principle If 𝑓 is a polynomial function and 𝑎 and 𝑏 are two numbers such that 𝑓 𝑎 < 0 and 𝑓 𝑏 > 0, then 𝑓 has at least one zero between 𝑎 and 𝑏. There must be a zero between 2 and 3 since the graph must cross the x-axis to connect these points. Location Principle Location Principle If 𝑓 is a polynomial function and 𝑎 and 𝑏 are two numbers such that 𝑓 𝑎 < 0 and 𝑓 𝑏 > 0, then 𝑓 has at least one zero between 𝑎 and 𝑏. So we could use a table of values with the Location Principle to narrow our rational zero choices. Just look for where the function-values change signs: + to – or – to +. Exercise 9 Find all real zeros of the function. f ( x) 24 x 4 38x3 191x 2 157 x 28 Exercise 10 The Rational Zero Theorem only helps in finding rational zeros. How would we go about finding the irrational zeros of a polynomial function? Once it comes down to an unfactorable quadratic, use the quadratic formula or completing the square to find the last two zeros. Exercise 11 Find all real zeros of the function. f ( x) 9 x 4 3x3 30 x 2 6 x 12 Exercise 12 Find all real zeros of the function. f ( x) 10 x 4 11x3 42 x 2 7 x 12 Find Rational Zeros, II Objectives: 1. To use a graph to help find the zeros of a polynomial function 2. To use the Location Principle to help find the zeros of a polynomial function