(1) Find all of the zeros of f.

Slides:



Advertisements
Similar presentations
Notes 6.6 Fundamental Theorem of Algebra
Advertisements

Rational Root Theorem.
Complex Numbers The imaginary number i is defined as so that Complex numbers are in the form a + bi where a is called the real part and bi is the imaginary.
SECTION 3.6 COMPLEX ZEROS; COMPLEX ZEROS; FUNDAMENTAL THEOREM OF ALGEBRA FUNDAMENTAL THEOREM OF ALGEBRA.
2.5 Zeros of Polynomial Functions
Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem.
EXAMPLE 2 Find the zeros of a polynomial function
EXAMPLE 2 Find all zeros of f (x) = x 5 – 4x 4 + 4x x 2 – 13x – 14. SOLUTION STEP 1 Find the rational zeros of f. Because f is a polynomial function.
Section 5.6 – Complex Zeros; Fundamental Theorem of Algebra Complex Numbers Standard form of a complex number is: a + bi. Every complex polynomial function.
Lesson 2.5 The Fundamental Theorem of Algebra. For f(x) where n > 0, there is at least one zero in the complex number system Complex → real and imaginary.
5.7 Apply the Fundamental Theorem of Algebra
Synthetic Division. This method is used to divide polynomials, one of which is a binomial of degree one.
 Find a polynomial with specified zeros.  For a polynomial function with integer coefficients, find the rational zeros and the other zeros, if possible.
The Rational Root Theorem.  Is a useful way to find your initial guess when you are trying to find the zeroes (roots) of the polynomial.  THIS IS JUST.
Chapter 3 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Zeros of Polynomial Functions.
7.5.1 Zeros of Polynomial Functions
OUTLINE  Homework (and homework questions)  Ask any review questions you want  Review long division, solve by factoring and graphing calculators  BREAK.
3.4 Zeros of Polynomial Functions. The Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, where n>0, then f has at least one zero in.
Zeros of Polynomials 2.5.
3.4 Zeros of Polynomial Functions Obj: Find x-intercepts of a polynomial Review of zeros A ZERO is an X-INTERCEPT Multiple Zeros the zeros are x = 5 (mult.
Section 3.3 Theorems about Zeros of Polynomial Functions.
Section 4.4 Theorems about Zeros of Polynomial Functions Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
Chapter 2 Polynomial and Rational Functions. Warm Up
7.3 Products and Factors of Polynomials Objectives: Multiply polynomials, and divide one polynomial by another by using long division and synthetic division.
2.5 The Fundamental Theorem of Algebra Students will use the fundamental theorem of algebra to determine the number of zeros of a polynomial. Students.
Remainder Theorem If f(x) is divided by x – r, then the remainder is equal to f(r). We can find f(r) using Synthetic Division.
Bellwork Perform the operation and write the result in standard from ( a + bi)
2015/16 TI-Smartview 2.5 The Fundamental Theorem of Algebra.
2.5 The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra – If f(x) is a polynomial of degree n, where.
Precalculus Lesson 2.5 The Fundamental Theorem of Algebra.
Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem.
Chapter 2 – Polynomial and Rational Functions 2.5 – The Fundamental Theorem of Algebra.
3.2 Division of Polynomials. Remember this? Synthetic Division 1. The divisor must be a binomial. 2. The divisor must be linear (degree = 1) 3. The.
Zeros (Solutions) Real Zeros Rational or Irrational Zeros Complex Zeros Complex Number and its Conjugate.
Remainder Theorem Let f(x) be an nth degree polynomial. If f(x) is divided by x – k, then the remainder is equal to f(k). We can find f(k) using Synthetic.
Chapter 6 Polynomials Review
Conjugate Pairs Let f (x) is a polynomial function that has real
Section 3.4 Zeros of Polynomial Functions
3.4 Zeros of Polynomial Functions
Notes Over 3.4 The Rational Zero Test
FST Chapter 7 Review Questions.
Warm - Up Perform the operation and write the result in standard form
3.8 Complex Zeros; Fundamental Theorem of Algebra
Lesson 2.5 The Fundamental Theorem of Algebra
Notes 5.6 (Day 1) Find Rational Zeros.
Section 3.4 Zeros of Polynomial Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Lesson 2.5 The Fundamental Theorem of Algebra
Zeros of a Polynomial Function
Today in Precalculus Go over homework Notes: Remainder
The Fundamental Theorem of Algebra
Fundamental Theorem of Algebra
Lesson: _____ Section 2.5 The Fundamental Theorem of Algebra
Warm-up: Find all real solutions of the equation X4 – 3x2 + 2 = 0
Rational Root Theorem.
The Factor Theorem A polynomial f(x) has a factor (x − k) if and only if f(k) = 0.
4.6 - Fundamental Theorem of Algebra
Divide. Use synthetic division.
Students, Take out your calendar and your homework
Solve the following equations.
(1) Find all of the zeros of f.
Students, Take out your calendar and your homework
Find all the real zeros of the functions.
Divide using long division.
1) Find f(g(x)) and g(f(x) to show that f(x) and g(x) are inverses
Multiply each expression.
Students, Take out your calendar and your homework
Remainder Theorem If P(x) is divided by x – r, then the remainder is equal to P(r). We can find P(r) using Synthetic Division.
Presentation transcript:

(1) Find all of the zeros of f. Students, Take out your calendar and your homework. Take out your spiral notebook and Complete the DNA. Use your notes if necessary. Use the function below to answer the questions. (1) Find all of the zeros of f. Without using long or synthetic division

3) Write the function as a product of its linear factors, and list all the zeros of f.

Ex 2) Find all the zeros of f(x) given that 2 – 2i is a zero of f.

Complex zeros occur in pairs If a + bi is a zero, then so is a – bi Ex 1) Find a fourth-degree polynomial function with real coefficients that has – 4, 3, and –2 – i as zeros.

4) Write as a product of linear factors: 5) Find a third-degree polynomial with integer coefficients that has 2 and 4 – i as zeros.

Write the following in standard form.

Find polynomial functions with the following zeros.