1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 3 Polynomial and Rational Functions

OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 The Real Zeros of a Polynomial Function SECTION Review real zeros of polynomials. Use the Rational Zeros Theorem. Find the possible number of positive and negative zeros of polynomials. Find the bounds on the real zeros of polynomials. Learn a procedure for finding the real zeros of a polynomial function. 4 5

3 © 2010 Pearson Education, Inc. All rights reserved REAL ZEROS OF A POLYNOMIAL FUNCTION 1. Definition If c is a real number in the domain of a function f and f(c) = 0, then c is called a real zero of f. a.Geometrically, this means that the graph of f has an x-intercept at x = c. b. Algebraically, this means that (x – c) is a factor of f(x).

4 © 2010 Pearson Education, Inc. All rights reserved REAL ZEROS OF A POLYNOMIAL FUNCTION 2. Number of real zeros A polynomial function of degree n has, at most, n real zeros. 3. Polynomial in factored form If a polynomial F(x) can be written in factored form, we find its zeros by solving F(x) = 0, using the zero product property of real numbers.

5 © 2010 Pearson Education, Inc. All rights reserved REAL ZEROS OF A POLYNOMIAL FUNCTION 4. Intermediate Value Theorem If a polynomial F(x) cannot be factored, we find approximate values of the real zeros (if any) of F(x) by using the Intermediate Value Theorem.

6 © 2010 Pearson Education, Inc. All rights reserved THE RATIONAL ZEROS THEOREM 1. p is a factor of the constant term a 0 ; 2. q is a factor of the leading coefficient a n. If is a polynomial function with integer coefficients (a n ≠ 0, a 0 ≠ 0) and is a rational number in lowest terms that is a zero of F(x), then

7 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Using the Rational Zeros Theorem Find all the rational zeros of List all possible zeros Solution

8 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Begin testing with 1. if it is not a rational zero, then try another possible zero. Solution continued The remainder of 0 tells us that (x – 1) is a factor of F(x). The other factor is 2x 2 + 7x + 3. Using the Rational Zeros Theorem

9 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solution continued The solution set is The rational zeros of F are Using the Rational Zeros Theorem

10 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Solving a Polynomial Equation Solve 3x 3 – 8x 2 – 8x + 8 = 0. Solution Possible rational roots

11 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Solving a Polynomial Equation Solution continued From the graph, it appears that an x-intercept is between 0.5 and 1.

12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Solving a Polynomial Equation Solution continued We check whether is a root by synthetic division. is a factor of the original equation with the depressed equation 3x 2 – 6x – 12 = 0.

13 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Solving a Polynomial Equation Solution continued So

14 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Solving a Polynomial Equation Solution continued

15 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Solving a Polynomial Equation Solution continued You can see from the graph that are also the x-intercepts of the graph. The three real roots of the given equation are

16 © 2010 Pearson Education, Inc. All rights reserved DESCARTE’S RULE OF SIGNS Let F(x) be a polynomial function with real coefficients and with terms written in descending order. 1.The number of positive zeros of F is either equal to the number of variations of sign of F(x) or less than that number by an even integer.

17 © 2010 Pearson Education, Inc. All rights reserved DESCARTE’S RULE OF SIGNS When using Descarte’s Rule, a zero of multiplicity m should be counted as m zeros. 2.The number of negative zeros of F is either equal to the number of variations of sign of F(–x) or less than that number by an even integer.

18 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Using Descarte’s Rule of Signs Find the possible number of positive and negative zeros of Solution There are three variations in sign in f (x). The number of positive zeros is either 3 or (3 – 2 =) 1.

19 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Using Descarte’s Rule of Signs Solution continued There are two variations in sign in f (–x). The number of negative zeros is either 2 or (2 – 2 =) 0.

20 © 2010 Pearson Education, Inc. All rights reserved RULES FOR BOUNDS Let F(x) be a polynomial function with real coefficients and a positive leading coefficient. Suppose F(x) is synthetically divided by x – k. Then 1.If k > 0, and each number in the last row is either zero or positive, then k is an upper bound on the zeros of F(x).

21 © 2010 Pearson Education, Inc. All rights reserved RULES FOR BOUNDS Zeros in the last row can be regarded as positive or negative. 2.If k < 0, and numbers in the last row alternate in sign, then k is a lower bound on the zeros of F(x).

22 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Bounds on the Zeros Find upper and lower bounds on the zeros of Solution The possible zeros are: ±1, ±2, ±3, and ±6. There is only one variation in sign so there is one positive zero. Use synthetic division with positive numbers until last row is all positive or 0. This first occurs for 3.

23 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Bounds on the Zeros Solution continued So 3 is upper bound. Use synthetic division with negative numbers until last row alternates in sign. This first occurs for –2. So –2 is lower bound.

24 © 2010 Pearson Education, Inc. All rights reserved FINDING THE REAL ZEROS OF A POLYNOMIAL Step 1 Find the maximum number of real zeros by using the degree of the polynomial function. Step 2 Find the possible number of positive and negative zeros by using Descartes’s Rule of Signs. Step 3 Write the set of possible rational zeros.

25 © 2010 Pearson Education, Inc. All rights reserved FINDING THE REAL ZEROS OF A POLYNOMIAL Step 4 Test the smallest positive integer in the set in Step 3, the next larger, and so on, until an integer zero or an upper bound of the zeros is found. a. If a zero is found, use the function represented by the depressed equation in further calculations. b. If an upper bound is found, discard all larger numbers in the set of possible rational zeros.

26 © 2010 Pearson Education, Inc. All rights reserved FINDING THE REAL ZEROS OF A POLYNOMIAL Step 5 Test the positive fractions that remain in the set in Step 3 after considering any bound that has been found. Step 6 Use modified Steps 4 and 5 for negative numbers in the set in Step 3. Step 7 If a depressed equation is quadratic, use any method (including the quadratic formula) to solve this equation.

27 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Finding the Real Zeros of a Polynomial Function Find the real zeros of Solution 1. f has, at most, 5 real zeros 2. There are three variations of sign in f(x). So f has 3 or 1 positive zeros.

28 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Finding the Real Zeros of a Polynomial Function Solution continued 2. continued There are two variations of signs in f(–x). So f has 2 or 0 negative zeros. 3. Possible rational zeros are

29 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Finding the Real Zeros of a Polynomial Function Solution continued 4. Test for zeros, 1, 2, 3, 4, 6, and 12, until a zero or an upper bound is found. (x – 2) is a factor of f, so

30 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Finding the Real Zeros of a Polynomial Function Solution continued 4. continued Consider The possible positive rational zeros of Q 1 (x) are Discard 1 since it is not a zero of the original function.

31 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Finding the Real Zeros of a Polynomial Function Solution continued 4. continued Test 2: 2 is not a zero, but is an upper bound.

32 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Finding the Real Zeros of a Polynomial Function Solution continued 5. Test remaining positive entries

33 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Finding the Real Zeros of a Polynomial Function Solution continued 5. continued Since is a zero,

34 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Finding the Real Zeros of a Polynomial Function Solution continued 6. Find the negative zeros of Q 2 (x) = x 3 + 2x 2 – 3x – 6. Test –1, –2, –3, and –6.

35 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Finding the Real Zeros of a Polynomial Function Solution continued 7. We can solve the depressed equation x 2 – 3 = 0 to obtain The real zeros of f are and 2. Because f(x) has, at most, five real zeros, we have found all of them.