If a polynomial q(x) is divided by x – 4, the quotient is 2

Slides:



Advertisements
Similar presentations
Notes 6.6 Fundamental Theorem of Algebra
Advertisements

Roots & Zeros of Polynomials I
SECTION 3.6 COMPLEX ZEROS; COMPLEX ZEROS; FUNDAMENTAL THEOREM OF ALGEBRA FUNDAMENTAL THEOREM OF ALGEBRA.
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.
The Fundamental Theorem of Algebra Section 4.6 beginning on page 198.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Fundamental Theorem of Algebra ♦ Perform arithmetic operations on complex.
Objectives Fundamental Theorem of Algebra 6-6
Lesson 4-1 Polynomial Functions.
9.9 The Fundamental Theorem of Algebra
7.5.1 Zeros of Polynomial Functions
6.6 The Fundamental Theorem of Algebra
7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.
Splash Screen.
General Results for Polynomial Equations In this section, we will state five general theorems about polynomial equations, some of which we have already.
Section 3.3 Theorems about Zeros of Polynomial Functions.
Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.
Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number.
1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 3 Polynomial and Rational Functions.
Essential Questions How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots?
Complex Zeros and the Fundamental Theorem of Algebra.
Objectives Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots. Identify all of the.
7.5 Roots and Zeros Objectives: The student will be able to…
Remainder and Factor Theorems
The Fundamental Theorem of Algebra It’s in Sec. 2.6a!!! Homework: p odd, all.
Fundamental Theorem of Algebra Every polynomial function of positive degree with complex coefficients has at least one complex zero.
Copyright © 2011 Pearson, Inc. 2.5 Complex Zeros and the Fundamental Theorem of Algebra.
5.6 The Fundamental Theorem of Algebra. If P(x) is a polynomial of degree n where n > 1, then P(x) = 0 has exactly n roots, including multiple and complex.
Section 4.6 Complex Zeros; Fundamental Theorem of Algebra.
Roots & Zeros of Polynomials II Finding the Solutions (Roots/Zeros) of Polynomials: The Fundamental Theorem of Algebra The Complex Conjugate Theorem.
Today in Pre-Calculus Notes: –Fundamental Theorem of Algebra –Complex Zeros Homework Go over quiz.
1 What you will learn today…  How to use the Fundamental Theorem of Algebra to determine the number of zeros of a polynomial function  How to use your.
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.
Section 6: Fundamental Theorem of Algebra Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with.
Solve polynomial equations with complex solutions by using the Fundamental Theorem of Algebra. 5-6 THE FUNDAMENTAL THEOREM OF ALGEBRA.
Every polynomial P(x) of degree n>0 has at least one zero in the complex number system. N Zeros Theorem Every polynomial P(x) of degree n>0 can be expressed.
Solving Polynomial Equations A-APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph.
Zeros of Polynomial Functions A-APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is.
Fundamental Theorem of Algebra
Fundamental Theorem of Algebra
Roots and Zeros 5.7.
FST Chapter 7 Review Questions.
Algebra II Explorations Review ( )
Complex Zeros and the Fundamental Theorem of Algebra
Solving Polynomial Functions
The Fundamental Theorem of Algebra
Fundamental Theorem of Algebra
Review & 6.6 The Fundamental Theorem of Algebra
7.5 Zeros of Polynomial Functions
Rational Root and Complex Conjugates Theorem
Finding Real Roots of Polynomial Equations
Lesson 7.2: Finding Complex Solutions of Polynomial Equations
7.5 Zeros of Polynomial Functions
Lesson 2.5 The Fundamental Theorem of Algebra
SECTION 9-3 : SOLVING QUADRATIC EQUATIONS
6-8 Roots and Zeros – Day 1 SWBAT
Today in Precalculus Go over homework Notes: Remainder
Apply the Fundamental Theorem of Algebra
The area of a rectangle is 2
Fundamental Theorem of Algebra
3.4 Zeros of Polynomial Functions: Real, Rational, and Complex
4.6 - Fundamental Theorem of Algebra
Fundamental Theorem of Algebra
Roots & Zeros of Polynomials I
Fundamental Theorem of Algebra
Fundamental Thm. Of Algebra
6-8 Roots and Zeros Given a polynomial function f(x), the following are all equivalent: c is a zero of the polynomial function f(x). x – c is a factor.
Problem of the Day Janice is solving the quadratic equation as shown:
5.8 Analyzing Graphs of Polynomials
Fundamental Theorem of Algebra
Presentation transcript:

If a polynomial q(x) is divided by x – 4, the quotient is 2 𝑥 2 +3𝑥+11+ 58 𝑥−4 . Which of the following must be true? q(-4) = 58 q(4) = 58 q(-4) does not exist q(4) does not exist Problem of the Day

Problem of the Day… Take 2! Given that p(-2) = 0, p(-5) = 0, and p(4) = 0, which expression could be p(x)? x 3 +3 x 2 −18x−40 x 3 −3 x 2 +18x−40 x 3 −3 x 2 −18x+40 x 3 −3 x 2 −18x−40 Problem of the Day… Take 2!

Section 5-7 Roots and Zeros

Then Now Objectives You used complex numbers to describe solutions of quadratic equations. Determine the number and type of roots for a polynomial equation. Find the zeros of a polynomial function.

Common Core State Standards Content Standards N.CN.9 – Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. A.APR.3 – Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Mathematical Practices 6) Attend to precision. Common Core State Standards

Zeros, Factors, Roots, and Intercepts

Find all of the zeros of: 𝑓 𝑥 = 𝑥 3 +2 𝑥 2 +9𝑥+18 Example 3

Find all of the zeros of: 𝑔 𝑥 = 𝑥 4 +50 𝑥 2 +49 Example 3

Find all of the zeros of: ℎ 𝑥 = 𝑥 4 −3 𝑥 3 − 𝑥 2 −27𝑥−90 Example 3

Find all of the zeros of: 𝑓 𝑥 = 𝑥 4 −3 𝑥 3 −9 𝑥 2 +77𝑥+150 Example 3

Complex Conjugates Theorem

Write a polynomial function of least degree with integral coefficients, the zeros of which include -5, 4i. Example 4

Write a polynomial function of least degree with integral coefficients, the zeros of which include 6, -2i. Example 4

Write a polynomial function of least degree with integral coefficients, the zeros of which include -1, 4, 3i. Example 4

p.363 #11, 15, 35, 38, 41, 42, 50 – 52, 61 Homework

Which of the following functions has only real zeros? a) b) c) d) Problem of the Day