7.5.1 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem to find the zeros of a polynomial function

Rational Root Theorem Let f(x) be a polynomial function with integer coefficients in standard form. If (in lowest terms) is a root of f(x) = 0, then p is a factor of the constant term of f(x) q is a factor of the leading coefficient of f(x)

Example 1 Find all rational roots of 8x3 + 10x2 – 11x + 2 = 0. Step 1: Make an organized list of all possible roots. factors of 2: factors of 8: List: ±1, ±½, ±¼, ± ⅛ , ±2

Example 1 Find all rational roots of 8x3 + 10x2 – 11x + 2 = 0. Step 2: Use substitution or synthetic division to test possible roots, until you find one that works. List: ±1, ±½, ±¼, ± ⅛ , ±2

Example 1 Find all rational roots of 8x3 + 10x2 – 11x + 2 = 0. Step 3: Use factoring or quadratic formula to find the other two. Found: ½ Resulting Equation: 8x2 + 14x – 4 = 0

Example 1 Find all rational roots of 8x3 + 10x2 – 11x + 2 = 0. Step 2: Use a graphing calculator to identify possible roots. possible roots: -2

Example 1 Find all rational roots of 8x3 + 10x2 – 11x + 2 = 0. Step 3: Use substitution or synthetic division to test all possible roots. possible roots: -2 -2 8 10 -11 2 -16 12 -2 8 -6 1 roots: -2

Example 1 Find all rational roots of 8x3 + 10x2 – 11x + 2 = 0. Step 3: Use substitution or synthetic division to test all possible roots. possible roots: -2 8 10 -11 2 2 3 -2 8 12 -8 roots -2

Example 1 Find all rational roots of 8x3 + 10x2 – 11x + 2 = 0. Step 3: Use substitution or synthetic division to test all possible roots. possible roots: -2 8 10 -11 2 4 7 -2 8 14 -4 roots: -2

Example 2 Find all of the zeros of Q(x) = x3 + 4x2 – 6x - 12. First, use the Rational Root Theorem and a graph of the polynomial function to determine some possibilities. Then use synthetic division to test your choices. 2 1 4 -6 -12 2 12 12 1 6 6 Since the remainder is 0, x – 2 is a factor of x3 + 4x2 – 6x - 12.

Example 2 Find all of the zeros of Q(x) = x3 + 4x2 – 6x - 12. or Since the remainder is 0, x – 2 is a factor of x3 + 4x2 – 6x - 12.

Homework p.463 #11-21 Odd

7.5.2 Zeros of Polynomial Functions Objectives: Use the Complex Conjugate Root Theorem to find the zeros of a polynomial function Use the Fundamental Theorem to write a polynomial function given sufficient information about its zeros

Example 1 Find all of the zeros of P(x) = -4x3 + 2x2 – x + 3. First, use the Rational Root Theorem and a graph of the polynomial function to determine some possibilities. Then use synthetic division to test your choices. 1 -4 2 -1 3 -4 -2 -3 -4 -2 -3 Since the remainder is 0, x – 1 is a factor of -4x3 + 2x2 – x + 3.

Example 1 Find all of the zeros of P(x) = -4x3 + 2x2 – x + 3. or Since the remainder is 0, x – 1 is a factor of -4x3 + 2x2 – x + 3.

Complex Conjugate Root Theorem If P is a polynomial function with real-number coefficients and a + bi (where b = 0) is a root of P(x) = 0, then a – bi is also a root of P(x) = 0. Graph R(x) = 2x3 – x2 – 4x. How many real zeros does R have? Graph S(x) = 2x3 – x2 – 4x + 3. How many real zeros does S have? Graph T(x) = 2x3 – x2 – 4x + 6. How many real zeros does T have?

Homework Page 464 #23-33 Odd

Example 2 Write a polynomial function, P, given that P has a degree of 3, P(0) = 120, and its zeros are -3, 2, and 4. Since the zeros are -3, 2, and 4, x = -3 x = 2 and x = 4 x + 3= 0 x - 2 = 0 and x – 4= 0 P(x) = a(x + 3)(x – 2)(x – 4) P(x) = a(x2 + x - 6)(x – 4) P(x) = a(x3 - 3x2 – 10x + 24) 120 = a((0)3 - 3(0)2 – 10(0) + 24) 120 = 24 a a = 5 P(x) = 5(x3 - 3x2 – 10x + 24) P(x) = 5x3 - 15x2 – 50 x + 120

Example 3 Write a polynomial function, P, given that P has a degree of 2, P(0) = 18, and its zeros are -3 + 3i and -3 – 3i. Since the zeros are -3 + 3i and -3 – 3i, x = -3 + 3i and x = -3 - 3i x – (-3 + 3i) = 0 and x – (-3 - 3i) = 0 x + 3 - 3i = 0 and x + 3 + 3i = 0 ( )( ) = 0 x + 3 - 3i x + 3 + 3i x2 + 3x + 3xi + 3x + 9 + 9i – 3xi – 9i – 9i2 = 0 x2 + 6x + 9 – 9(-1) = 0 x2 + 6x + 18 = 0

Homework Page 464 #41-49 Odd