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Warm Up • • • • • Find the degree and leading coefficient 1. 𝑥 3 + 3𝑥 2 + 16𝑥 + 48 2. −4𝑥 7 + 2 3. −𝑥 4 − 5𝑥 2 4. 5𝑥 − 3 Algebra 3 Chapter 6 Lesson 7: Using the Fundamental Theorem of Algebra VOCAB • The Fundamental Theorem of Algebra – if 𝑓(𝑥) is a polynomial of degree n where n > 0 then 𝑓 𝑥 = 0 has at least one root in the set of complex numbers • Repeated Solution – is when you have the same answer twice (this comes from a square) – Ex (𝑥 − 2)2 GOAL • Today you will be able to find how many total zeros a problem has. – This is different than last week. – Last week was REAL zeros • Today will include imaginary Directions (Number of Solutions/ Zeros) • Write the problem in standard form • Look at the degree (that’s your answer) I DO (Number of Solutions/ Zeros) • Find how many solutions/zeros the polynomial has. • 1. 𝑥 3 + 3𝑥 2 + 16𝑥 + 48 = 0 • 2. 𝑥 4 + 6𝑥 3 + 12𝑥 2 + 8𝑥 • 3. 𝑥 3 − 3𝑥 2 + 4𝑥 + 2 = 0 • 4. 5𝑥 − 7 WE DO (Number of Solutions/ Zeros) • Find how many solutions/zeros the polynomial has. • 1. 𝑥 19 − 2𝑥 4 − 2 = 0 • 2. −𝑥 4 • 3. 7𝑥 21 + 13𝑥 12 − 32 = 0 • 4. 4 YOU DO (Number of Solutions/ Zeros) • Find how many solutions/zeros the polynomial has. • 1. −2𝑥 31 + 7𝑥 24 − 13𝑥 + 4 = 0 • 2. 9𝑥 164 + 64𝑥 314 + 13𝑥 2128 + 8𝑥 • 3. -3𝑥 123 − 3𝑥 456 + 4𝑥 = 0 • 4. 17𝑥 REVIEW • What did you just learn? Homework • Worksheet – 6.7B (1 – 4) Directions (Identifying Factors) • Numbers – Take the opposite – Write (x ____ ) • Complex Numbers – You need to take both • 𝑎 + 𝑏𝑖 • 𝑎 − 𝑏𝑖 – Write (x –(𝑎 + 𝑏𝑖)) or (x –(𝑎 − 𝑏𝑖)) I DO (Identifying Factors) • Identify the factors of a polynomial function that has the given zeros • 1. 3, 0, -2 • 2. 1, 1, i, -i • 3. 5, 2 + 3i • 4. 1, -1, 2, -2, 3 WE DO (Identifying Factors) • Identify the factors of a polynomial function that has the given zeros • 1. 3, -2, -1 + i • 2. 4i, 4i • 3. -4, 5, 5 • 4. -3, 4i, -4i Warm Up • Find how many solutions/zeros the polynomial has. • 1. 13𝑥 + 4 = 0 • 2. 9𝑥 14 + 64𝑥 34 + 13𝑥 79 + 8𝑥 • 3. -4𝑥 215 + 3𝑥 251 − 7 = 0 • 4. 17 TODAY • We will learn how to identify factors of a polynomial and write a polynomial function from the zeros WE DO (Identifying Factors) • Identify the factors of a polynomial function that has the given zeros • 1. -2, -2, -2, 0 • 2. 6 • 3. 3, 5 - i • 4. -4, -4, 2i, -2i YOU DO (Identifying Factors) • Identify the factors of a polynomial function that has the given zeros • 1. 3, 0, -i • 2. 5, 5, -4 - i • 3. -6, 2, 1, 1 • 4. 5i, 2, 3 - 2i REVIEW • Today we learned how to identify the factors of a polynomial function when given the zeros HOMEWORK • Worksheet – 6.7B (9 - 14) DIRECTIONS (Writing Polynomials) • Identify the factors • Simplify – FOIL – Combine Like Terms Warm Up • Identify the factors of a polynomial function that has the given zeros • 1. 2, -3, 4i, 5 - i I DO (Writing Polynomials) • Write a polynomial function of least degree that has real coefficients, the given zeros, and a leading coefficient of 1 • 1. 6 • 2. -3 , 5 • 3. 3 + 4i, 0 • 4. 2 – i, 1 , 2 WE DO (Writing Polynomials) • Write a polynomial function of least degree that has real coefficients, the given zeros, and a leading coefficient of 1 • 1. -1, 1, 3 • 2. 2, 3, 0 • 3. 5 -2i, i, 0, 3 • 4. 2i, -2i, 5i, -5i YOU DO (Writing Polynomials) • Write a polynomial function of least degree that has real coefficients, the given zeros, and a leading coefficient of 1 • 1. 0, 3, -4 • 2. -2, 1 • 3. 4 + i, 4 – i, -2 • 4. 3i, 2, -4 Review • What did you learn? Homework • Worksheet – 6.7B (15 – 23) WARM UP •DO NOT TURN IN YOUR HOMEWORK •Write the equation • 1. 2, 0, 3, -1 TODAY • We will review writing polynomial equations when given the zeros Remember • Quiz Tomorrow • Make a cheat sheet