Chapter 6 Review.

Slides:

Advertisements

Similar presentations

The Rational Zero Theorem

Advertisements

Lesson 3.4 – Zeros of Polynomial Functions Rational Zero Theorem

Polynomials Identify Monomials and their Degree

Chapter 6 – Polynomial Functions

Creating Polynomials Given the Zeros.. What do we already know about polynomial functions? They are either ODD functions They are either EVEN functions.

Zeros of Polynomial Functions

Pre-Calculus For our Polynomial Function: The Factors are:(x + 5) & (x - 3) The Roots/Solutions are:x = -5 and 3 The Zeros are at:(-5, 0) and (3, 0)

Dividing Polynomials.

4.1 Polynomial Functions Objectives: Define a polynomial.

The Rational Zero Theorem

The Fundamental Theorem of Algebra And Zeros of Polynomials

Polynomial Functions Chapter 2 Part 1. Standard Form f(x)=ax 2 +bx+c Vertex Form f(x)=a(x-h) 2 +k Intercept Form f(x)=a(x-d)(x-e) y-int (0, c) let x =

6.1 Using Properties of Exponents What you should learn: Goal1 Goal2 Use properties of exponents to evaluate and simplify expressions involving powers.

Polynomial Functions and Inequalities

Algebra 2 Chapter 6 Notes Polynomials and Polynomial Functions Algebra 2 Chapter 6 Notes Polynomials and Polynomial Functions.

4-5, 4-6 Factor and Remainder Theorems r is an x intercept of the graph of the function If r is a real number that is a zero of a function then x = r.

Polynomial Functions and Inequalities

Algebra 2.  Warm Up  A monomial is an expression that is either a real number, a variable or a product of real numbers and variables.  A polynomial.

Lesson 3.4 – Zeros of Polynomial Functions Rational Zero Theorem

Multiply polynomials vertically and horizontally

Polynomials Identify monomials and their degree Identify polynomials and their degree Adding and Subtracting polynomial expressions Multiplying polynomial.

Zeros of Polynomials 2.5.

The Remainder Theorem A-APR 2 Explain how to solve a polynomial by factoring.

1. Describe the end behavior of the graph y = 2x 5 – 3x Sketch a graph of 3 rd degree with a zero at -5 (multiplicity 2) and a zero at 0 (multiplicity.

The Rational Zero Theorem The Rational Zero Theorem gives a list of possible rational zeros of a polynomial function. Equivalently, the theorem gives all.

Real Zeros of Polynomials Section 2.4. Review – Long Division 1. What do I multiply by to get the first term? 2. Multiply through 3. Subtract 4. Bring.

AFM Objective  Polynomials have more than one term: ◦ 2x + 1 ◦ 3x 2 + 2x + 1  The fancy explanation: ◦ f(x) = a n x n + a n-1 x n-1 + …+ a 2 x.

6-3 DIVIDING POLYNOMIALS Synthetic division. Using synthetic division to perform long division of polynomials  Procedures to follow for synthetic division:

Quadratic Function A quadratic function is defined by a quadratic or second-degree polynomial. Standard Form

Chapter 11 Polynomial Functions

3.3 Dividing Polynomials.

Polynomial Functions and their Graphs

Today in Pre-Calculus Go over homework Notes: (need calculator & book)

LESSON 2–2 Polynomial Functions.

How do you write an equation in Scientific Notation?

When given a root and when not given a root

Polynomial Long Division Review

Lesson 7.4 Solving polynomial equations in factored form

Warm-up Solve each equation 4+13= −3+7= 4−13= −3−7=

Polynomials and Polynomial Functions

Dividing Polynomials.

Warm Up: Solve & Sketch the graph:

Mod. 3 Day 4 Graphs of Functions.

The Rational Zero Theorem

Polynomials November 28 Tuesday.

4.1 Objective: Students will look at polynomial functions of degree greater than 2, approximate the zeros, and interpret graphs.

Precalculus PreAP/Dual, Revised ©2017

By: Deanna Carbone, Jacqueline DiSalvatore, and Alyssa Fanelli

Apply the Fundamental Theorem of Algebra Lesson 2.7

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Creating Polynomials Given the Zeros.

Introduction Two equations that are solved together are called systems of equations. The solution to a system of equations is the point or points that.

Unit 4 Polynomials.

Warm Up Graph the following y= x4 (x + 3)2 (x – 4)5 (x – 3)

Algebra II – Pre-requisite Skills Mr. Rosilez

5.5 - Long and Synthetic Division

Warm Up Graph the following y= x4 (x + 3)2 (x – 4)5 (x – 3)

Algebra 1 Chapters 7-10.

The Fundamental Theorem of Algebra And Zeros of Polynomials

Polynomial Functions and Their Graphs

The Rational Zero Theorem

Rational Root Theorem.

Dividing Polynomials WOW! I want to learn how to do that?

Page ) Origin 21) 23) Y-axis, Origin 25) Odd 27) Even 29) 31)

Chapter 2 Section 2 Linear Equations.

Polynomial Functions Unit 5 Algebra 2A.

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.

Review of Integers and Solving Equations

Inverse Trigonometric Functions: Integration

Presentation transcript:

Chapter 6 Review

Main Concepts Properties of Exponents Determining End Behavior of Polynomial Functions Adding, Subtracting, and Multiplying Polynomials Finding Zeros of Polynomial Functions Descartes’ Rule of Signs Making P/Q values Long and synthetic division Graphing Polynomial Functions End behavior, multiplicity rule Factor by Grouping and u substitution

Properties of Exponents 2 2

End Behavior End behavior is how each polynomial behaves as it approaches ∞ or -∞ . The -2 is the value that determines whether the graph is positive or negative, while the exponent (5) is the value that determines whether the arms of the graph will go the same or different ways. In this case, the graph is negative because -2 is negative, and the arms go opposite ways because the exponent is odd. If the exponent had been even, both arms would face the same way.

Coefficient is pos. Coefficient is neg. Coefficient is pos Coefficient is pos. Coefficient is neg. Coefficient is pos. Coefficient is neg. Exponent is even Exponent is even Exponent is odd Exponent is odd

Adding, Subtracting, Multiplying Polynomials Add: (x + 3) + (2x - 4) Answer: 3x - 1 Subtract: (3x - 7) - (x - 1) Answer: 2x - 6 Multiply: (3x - 1)(x - 4)(x + 1) Answer: (see below)

Factoring by grouping, u substitution To factor by grouping, find a common denominator comprised of two terms in a four-term equation. For example, can be factored by grouping first into x^2(x-2) + 4(x-2), then finally into (x^2+4)(x-2). To u substitute, replace with u and then solve like a quadratic. For example, can be substituted into u^2 +3u + 2, then into (u+2)(u+1), then back into (x^2+2)(x^2+1).

Synthetic Division tested zero (n) leading coefficient (a) coefficient of second term (b) coefficient of third term (c) coefficient of fourth term (d) a * n (b + (a * n)) * n ... a b + (a * n) ((b + (a*n))*n) + c

Ex. Determine whether or not x = 2 is a zero of the following polynomial:

Polynomial Long Division

Descartes’ Rule of Signs Descartes’ Rule of Signs states that: The number of positive solutions of a polynomial is equal to the number of sign changes in f(x) or the number of sign changes minus an even number. (ex. If there are 7 sign changes, there are either 7, 5, 3, or 1 solution(s)) The number of negative solutions of a polynomial is equal to the number of sign changes in f(-x) or the number of sign changes minus an even number. The other solutions are imaginary.

+ - i 3 1 2 Sign changes: 3 Sign changes: 1

P/Q Values P Q Divisor of P / Divisor of Q = Possible polynomial solution.

Multiplicity Rule Multiplicity is when a zero occurs more than once in a polynomial. If a zero is repeated an odd number of times, the graph will cross the X-Axis at that zero. If a zero is repeated an even number of times, the graph will not cross the X-axis at that zero and rather become a turning point.

Put these concepts together! Graph:

Jeopardy: https://www.playfactile.com/chaptersixmath/play