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Math Lesson2.notebook October 31, 2014 5.6 Quiz Find all real zeros of the function. 1. f(x) = x3 12x2 + 35x 24 1 Math Lesson2.notebook October 31, 2014 Today's Objective ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Students will learn how to identify rational, irrational, and imaginary zeros of polynomials by using the fundamental theorem of algebra. 2 Math Lesson2.notebook 5.7 Apply the Fundamental Theorem of Algebra Day 1 October 31, 2014 10/31 Fundamental Theorem of Algebra: If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one solution in the set of complex numbers. ***If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has exactly n solutions provided each solution repeated twice is counted at 2 solutions... Ex. 1: Find the number of solutions or zeros. a. x3 + 5x2 + 4x + 20 = 0 b. f(x) = x4 8x3 + 18x2 27 3 Math Lesson2.notebook October 31, 2014 You Practice 1. How many solutions does the equation have? x4 + 5x2 36 = 0 2. How many zeros does the function have? f(x) = x3 + 7x2 + 8x 16 4 Math Lesson2.notebook October 31, 2014 Ex. 2: Find all zeros of f(x) = x5 4x4 + 4x3 + 10x2 13x 14 step 1 Find the rational zeros of f. step 2 Write f(x) in factored form. step 3 Find the complex zeros of f. 5 Math Lesson2.notebook October 31, 2014 You Practice Find all zeros of the polynomial function. 3. f(x) = x3 + 7x2 + 15x + 9 4. f(x) = x5 2x4 + 8x2 13x + 6 6 Math Lesson2.notebook October 31, 2014 Complex Conjugates Theorem: If f is a polynomial function with real coefficients, and a + bi is an imaginary zero of f, then a bi is also a zero of f. Irrational Conjugates Theorem: Suppose f is a polynomial function with rational coefficients, and a and b are rational numbers such that √b is irrational. If a + √b is a zero, then a √b is also a zero. 7 Math Lesson2.notebook October 31, 2014 Ex. 3: Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and 3 and 2 + √5 as zeros. 8 Math Lesson2.notebook October 31, 2014 You 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 6. 4, 1 + √5 7. 2, 2i, 4 √6 8. 3, 3 i 9 Math Lesson2.notebook October 31, 2014 10
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