1. Warm-Up Exercises 1. What is the degree of f (x) = 8x 6 – 4x 5 + 3x 2 + 2 ? 2. Solve x 2 – 2x + 3 = 0 ANSWER 6 1 i 2 + _

2. Warm-Up Exercises 3. The function P given by x 4 + 3x 3 – 30x 2 – 6x = 56 model the profit of a company. What are the real solution of the function? ANSWER –7, 2, 4 + _

3. Warm-Up Exercises EXAMPLE 1 Find the number of solutions or zeros a. How many solutions does the equation x 3 + 5x 2 + 4x + 20 = 0 have? SOLUTION Because x 3 + 5x 2 + 4x + 20 = 0 is a polynomial equation of degree 3,it has three solutions. (The solutions are – 5, – 2i, and 2i.)

4. Warm-Up Exercises EXAMPLE 1 Find the number of solutions or zeros b. How many zeros does the function f (x) = x 4 – 8x 3 + 18x 2 – 27 have? SOLUTION Because f (x) = x 4 – 8x 3 + 18x 2 – 27 is a polynomial function of degree 4, it has four zeros. (The zeros are – 1, 3, 3, and 3.)

5. Warm-Up Exercises GUIDED PRACTICE for Example 1 1. How many solutions does the equation x 4 + 5x 2 – 36 = 0 have? ANSWER 4

6. Warm-Up Exercises GUIDED PRACTICE for Example 1 2. How many zeros does the function f (x) = x 3 + 7x 2 + 8x – 16 have? ANSWER 3

7. Warm-Up Exercises EXAMPLE 2 Find all zeros of f (x) = x 5 – 4x 4 + 4x 3 + 10x 2 – 13x – 14. SOLUTION STEP 1 Find the rational zeros of f. Because f is a polynomial function of degree 5, it has 5 zeros. The possible rational zeros are + 1, + 2, + 7, and + 14. Using synthetic division, you can determine that – 1 is a zero repeated twice and 2 is also a zero. STEP 2Write f (x) in factored form. Dividing f (x) by its known factors x + 1, x + 1, and x – 2 gives a quotient of x 2 – 4x + 7. Therefore: f (x) = (x + 1) 2 (x – 2)(x 2 – 4x + 7) Find the zeros of a polynomial function

8. Warm-Up Exercises EXAMPLE 2 STEP 3 Find the complex zeros of f. Use the quadratic formula to factor the trinomial into linear factors. f(x) = (x + 1) 2 (x – 2) x – (2 + i 3 ) x – (2 – i 3 ) The zeros of f are – 1, – 1, 2, 2 + i 3, and 2 – i 3. ANSWER Find the zeros of a polynomial function

9. Warm-Up Exercises GUIDED PRACTICE for Example 2 Find all zeros of the polynomial function. 3. f (x) = x 3 + 7x 2 + 15x + 9 STEP 1 Find the rational zero of f. because f is a polynomial function degree 3, it has 3 zero. The possible rational zeros are 1, 3, using synthetic division, you can determine that  3 is a zero reputed twice and –3 is also a zero + – + – STEP 2 Write f (x) in factored form Formula are (x +1) 2 (x +3) f(x) = (x +1) (x +3) 2 The zeros of f are – 1 and – 3 SOLUTION

10. Warm-Up Exercises GUIDED PRACTICE for Example 2 4. f (x) = x 5 – 2x 4 + 8x 2 – 13x + 6 STEP 1 Find the rational zero of f. because f is a polynomial function degree 5, it has 5 zero. The possible rational zeros are 1, 2, 3 and Using synthetic division, you can determine that  1 is a zero reputed twice and –3 is also a zero + – + – + – + – 6. STEP 2 Write f (x) in factored form dividing f(x) by its known factor (x – 1),(x – 1) and (x+2) given a qualities x 2 – 2x +3 therefore f (x) = (x – 1) 2 (x+2) (x 2 – 2x + 3) SOLUTION

11. Warm-Up Exercises GUIDED PRACTICE for Example 2 Find the complex zero of f. use the quadratic formula to factor the trinomial into linear factor STEP 3 f (x) = (x –1) 2 (x + 2) [x – (1 + i 2) [ (x – (1 – i 2)] Zeros of f are 1, 1, – 2, 1 + i 2, and 1 – i 2

12. Warm-Up Exercises EXAMPLE 3 Use zeros to write a polynomial function Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and 3 and 2 + 5 as zeros. SOLUTION Because the coefficients are rational and 2 + 5 is a zero, 2 – 5 must also be a zero by the irrational conjugates theorem. Use the three zeros and the factor theorem to write f (x) as a product of three factors.

13. Warm-Up Exercises EXAMPLE 3 Use zeros to write a polynomial function f (x) = (x – 3) [ x – (2 + √ 5 ) ] [ x – (2 – √ 5 ) ] Write f (x) in factored form. Regroup terms. = (x – 3) [ (x – 2) – √ 5 ] [ (x – 2) + √ 5 ] = (x – 3)[(x – 2) 2 – 5] Multiply. = (x – 3)[(x 2 – 4x + 4) – 5] Expand binomial. = (x – 3)(x 2 – 4x – 1) Simplify. = x 3 – 4x 2 – x – 3x 2 + 12x + 3 Multiply. = x 3 – 7x 2 + 11x + 3 Combine like terms.

14. Warm-Up Exercises EXAMPLE 3 Use zeros to write a polynomial function f(3) = 3 3 – 7(3) 2 + 11(3) + 3 = 27 – 63 + 33 + 3 = 0  f(2 + √ 5 ) = (2 + √ 5 ) 3 – 7(2 + √ 5 ) 2 + 11( 2 + √ 5 ) + 3 = 38 + 17 √ 5 – 63 – 28 √ 5 + 22 + 11 √ 5 + 3 = 0  Since f (2 + √ 5 ) = 0, by the irrational conjugates theorem f (2 – √ 5 ) = 0.  You can check this result by evaluating f (x) at each of its three zeros. CHECK

15. Warm-Up Exercises = (x + 1) (x 2 – 4x – 2x + 8) f (x) = (x + 1) (x – 2) ( x – 4) GUIDED PRACTICE Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. 5. – 1, 2, 4 for Example 3 Write f (x) in factored form. = (x + 1) (x 2 – 6x + 8) Multiply. = x 3 – 6x 2 + 8x + x 2 – 6x + 8 = x3 – 5x 2 + 2x + 8 Multiply. Combine like terms. Use the three zeros and the factor theorem to write f(x) as a product of three factors. SOLUTION

16. Warm-Up Exercises GUIDED PRACTICE for Example 3 6. 4, 1 + √ 5 f (x) = (x – 4) [ x – (1 + √ 5 ) ] [ x – (1 – √ 5 ) ] Write f (x) in factored form. Regroup terms. = (x – 4) [ (x – 1) – √ 5 ] [ (x – 1) + √ 5 ] Multiply. = (x – 4)[(x 2 – 2x + 1) – 5] Expand binomial. = (x – 4)[(x – 1) 2 – ( 5) 2 ] Because the coefficients are rational and 1 + 5 is a zero, 1 – 5 must also be a zero by the irrational conjugates theorem. Use the three zeros and the factor theorem to write f (x) as a product of three factors SOLUTION

17. Warm-Up Exercises GUIDED PRACTICE for Example 3 = (x – 4)(x 2 – 2x – 4) Simplify. = x 3 – 2x 2 – 4x – 4x 2 + 8x + 16 Multiply. = x 3 – 6x 2 + 4x +16 Combine like terms.

18. Warm-Up Exercises GUIDED PRACTICE 7. 2, 2i, 4 – √ 6√ 6 for Example 3 Because the coefficients are rational and 2i is a zero, –2i must also be a zero by the complex conjugates theorem. 4 + 6 is also a zero by the irrational conjugate theorem. Use the five zeros and the factor theorem to write f(x) as a product of five factors. f (x) = (x–2) (x +2i)(x-2i)[(x –(4 – √ 6 )][x –(4+ √ 6) ] Write f (x) in factored form. Regroup terms. = (x – 2) [ (x 2 –(2i) 2 ][x 2 –4)+ √ 6][(x– 4) – √ 6 ] Multiply. = (x – 2)(x 2 + 4)(x 2 – 8x+16 – 6) Expand binomial. = (x – 2)[(x 2 + 4)[(x– 4) 2 – ( 6 ) 2 ] SOLUTION

19. Warm-Up Exercises GUIDED PRACTICE for Example 3 = (x – 2)(x 2 + 4)(x 2 – 8x + 10) Simplify. = (x–2) (x 4 – 8x 2 +10x 2 +4x 2 –3x +40) Multiply. = (x–2) (x 4 – 8x 3 +14x 2 –32x + 40) Combine like terms. = x 5 – 8x 4 +14x 3 –32x 2 +40x – 2x 4 +16x 3 –28x 2 + 64x – 80 = x 5 –10x 4 + 30x 3 – 60x 2 +10x – 80 Combine like terms. Multiply.

20. Warm-Up Exercises GUIDED PRACTICE 8. 3, 3 – i for Example 3 Because the coefficients are rational and 3 –i is a zero, 3 + i must also be a zero by the complex conjugates theorem. Use the three zeros and the factor theorem to write f(x) as a product of three factors = f(x) =(x – 3)[x – (3 – i)][x –(3 + i)] = (x–3)[(x– 3)+i ][(x 2 – 3) – i] Regroup terms. = (x–3)[(x – 3) 2 –i 2 )] = (x– 3)[(x – 3)+ i][(x –3) –i] Multiply. Write f (x) in factored form. SOLUTION

21. Warm-Up Exercises GUIDED PRACTICE for Example 3 = (x – 3)[(x – 3) 2 – i 2 ]=(x –3)(x 2 – 6x + 9) Simplify. = (x–3)(x 2 – 6x + 9) = x 3 –6x 2 + 9x – 3x 2 +18x – 27 Combine like terms. = x 3 – 9x 2 + 27x –27 Multiply.

22. Warm-Up Exercises EXAMPLE 4 Use Descartes’ rule of signs Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for f (x) = x 6 – 2x 5 + 3x 4 – 10x 3 – 6x 2 – 8x – 8. The coefficients in f (x) have 3 sign changes, so f has 3 or 1 positive real zero(s). f ( – x) = ( – x) 6 – 2( – x) 5 + 3( – x) 4 – 10( – x) 3 – 6( – x) 2 – 8( – x) – 8 = x 6 + 2x 5 + 3x 4 + 10x 3 – 6x 2 + 8x – 8 SOLUTION

23. Warm-Up Exercises EXAMPLE 4 Use Descartes’ rule of signs The coefficients in f (– x) have 3 sign changes, so f has 3 or 1 negative real zero(s). The possible numbers of zeros for f are summarized in the table below.

24. Warm-Up Exercises GUIDED PRACTICE for Example 4 Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for the function. 9. f (x) = x 3 + 2x – 11 The coefficients in f (x) have 1 sign changes, so f has 1 positive real zero(s). SOLUTION f (x) = x 3 + 2x – 11

25. Warm-Up Exercises GUIDED PRACTICE for Example 4 The coefficients in f (– x) have no sign changes. The possible numbers of zeros for f are summarized in the table below. f ( – x) = ( – x) 3 + 2(– x) – 11 = – x 3 – 2x – 11

26. Warm-Up Exercises GUIDED PRACTICE for Example 4 10. g(x) = 2x 4 – 8x 3 + 6x 2 – 3x + 1 The coefficients in f (x) have 4 sign changes, so f has 4 positive real zero(s). f ( – x) = 2( – x) 4 – 8(– x) 3 + 6(– x) 2 + 1 = 2x 4 + 8x + 6x 2 + 1 SOLUTION f (x) = 2x 4 – 8x 3 + 6x 2 – 3x + 1 The coefficients in f (– x) have no sign changes.

27. Warm-Up Exercises GUIDED PRACTICE for Example 4 The possible numbers of zeros for f are summarized in the table below.

28. Warm-Up Exercises EXAMPLE 5 Approximate real zeros Approximate the real zeros of f (x) = x 6 – 2x 5 + 3x 4 – 10x 3 – 6x 2 – 8x – 8. SOLUTION Use the zero (or root) feature of a graphing calculator, as shown below. From these screens, you can see that the zeros are x ≈ – 0.73 and x ≈ 2.73. ANSWER

29. Warm-Up Exercises EXAMPLE 6 Approximate real zeros of a polynomial model s (x) = 0.00547x 3 – 0.225x 2 + 3.62x – 11.0 What is the tachometer reading when the boat travels 15 miles per hour? A tachometer measures the speed (in revolutions per minute, or RPMs ) at which an engine shaft rotates. For a certain boat, the speed x of the engine shaft (in 100s of RPMs ) and the speed s of the boat (in miles per hour) are modeled by TACHOMETER

30. Warm-Up Exercises EXAMPLE 6 Approximate real zeros of a polynomial model Substitute 15 for s(x) in the given function. You can rewrite the resulting equation as: 0 = 0.00547x 3 – 0.225x 2 + 3.62x – 26.0 Then, use a graphing calculator to approximate the real zeros of f (x) = 0.00547x 3 – 0.225x 2 + 3.62x – 26.0. SOLUTION From the graph, there is one real zero: x ≈ 19.9. The tachometer reading is about 1990 RPMs.ANSWER

31. Warm-Up Exercises GUIDED PRACTICE for Examples 5 and 6 11. Approximate the real zeros of f (x) = 3x 5 + 2x 4 – 8x 3 + 4x 2 – x – 1. The zeros are x ≈ – 2.2, x ≈ – 0.3, and x ≈ 1.1.ANSWER

32. Warm-Up Exercises GUIDED PRACTICE for Examples 5 and 6 12. What If? In Example 6, what is the tachometer reading when the boat travels 20 miles per hour? Substitute 20 for s(x) in the given function. You can rewrite the resulting equation as: 0 = 0.00547x 3 – 0.225x 2 + 3.62x – 31.0 Then, use a graphing calculator to approximate the real zeros of f (x) = 0.00547x 3 – 0.225x 2 + 3.62x – 31.0. SOLUTION From the graph, there is one real zero: x ≈ 23.1.

33. Warm-Up Exercises GUIDED PRACTICE for Examples 5 and 6 The tachometer reading is about 2310 RPMs.ANSWER

34. Warm-Up Exercises Daily Homework Quiz 1. Find all the zeros of f(x) = x 4 – x 2 – 20. 2. Write a polynomial function of least degree that has rational coefficients, a leading coefficient of 1, and – 3 and 1 – 7i ANSWER x 3 + x 2 + 44x + 150. + √5, + 2i ANSWER

35. Warm-Up Exercises Daily Homework Quiz 3. Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for f(x) = 2x 5 – 3x 4 – 5x 3 + 10x 2 + 3x – 5. 3 positive, 2 negative, 0 imaginary; 3 positive, 0 negative, 2 imaginary; 1 positive, 2 negative, 2 imaginary; 1 positive, 0 negative, 4 imaginary ANSWER

36. Warm-Up Exercises 4. The profit P for printing envelopes is modeled by P = x – 0.001x 3 – 0.06x 2 + 30.5x, where x is the number of envelopes printed in thousands. What is the least number of envelopes that can be printed for a profit of $1500? Daily Homework Quiz ANSWER about 70,000 envelopes