6.2 Polynomials and Linear Functions

Slides:

Advertisements

Similar presentations

Rational Root Theorem.

Advertisements

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

Do Now Factor completely and solve. x2 - 15x + 50 = 0

Solving Polynomial Equations

Finding the Roots of a Polynomial. 2 Real Roots and 2 Complex Roots because it only has 2 x-intercepts but has 4 “turns” Classifying the Roots of a Polynomial.

Solving Quadratic Equation by Graphing Section 6.1.

Solving Quadratic Equations Tammy Wallace Varina High.

Roots & Zeros of Polynomials I How the roots, solutions, zeros, x-intercepts and factors of a polynomial function are related.

4.1 Polynomial Functions Objectives: Define a polynomial.

Warm up Use the Rational Root Theorem to determine the Roots of : x³ – 5x² + 8x – 6 = 0.

Polynomial Division: Dividing one polynomial by another polynomial to find the zeros of the polynomial. Ex 1: Find the zeros of Solution:1 st way: At.

1.3 “Solving Linear Equations” Steps: 1.Isolate the variable. 2.To solve when there is a fraction next to a variable, multiply both sides by the reciprocal.

Polynomials and Polynomial Functions

1. Solve: 2x 3 – 4x 2 – 6x = 0. (Check with GUT) 2. Solve algebraically or graphically: x 2 – 2x – 15> 0 1.

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

Polynomial Functions. A Polynomial is an expression that is either a real number, a variable, or a product of real numbers and variables with whole number.

Exponents An exponent is the number of times the base is multiplied by itself. Example 27 can also be written as 3 This means 3 X 3 X 3.

Notes - Solving Quadratic Equations in Factored Form If ab = 0, then a = 0 or b = 0 If the product of two factors is zero, then at least one of the factors.

Warm-up Given these solutions below: write the equation of the polynomial: 1. {-1, 2, ½)

Solving Quadratic Equations

Math is about to get imaginary!

Lesson 3.4 – Zeros of Polynomial Functions Rational Zero Theorem

Long Division Algorithm and Synthetic Division!!!

Factors, Remainders, and Roots, Oh My! 1 November 2010.

Warm Up Foil (3x+7)(x-1) Factors, Roots and Zeros.

 Carl Gauss provided a proof of the fundamental theorem of algebra at the age of 22.  Gauss is considered the prince of mathematics.  Gauss was able.

The Quadratic Formula & Discriminant Essential question – How do you solve a quadratic equation using the Quadratic Formula?

Polynomials, Factors and Zeros

Solving polynomial equations

6-2 Polynomials and Linear Factors. Standard and Factored Form  Standard form means to write it as a simplified (multiplied out) polynomial starting.

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

THE FUNDAMENTAL THEOREM OF ALGEBRA. Descartes’ Rule of Signs If f(x) is a polynomial function with real coefficients, then *The number of positive real.

5.2 Polynomials, Linear Factors, and Zeros P

Test info for chp 10 test – next week 30 problems in total Factoring – 8 Completing the square – 3 Multiplying polynomials – 9 Solve for “x” – 4 Adding/subtracting.

6.4 Solving Polynomial Equations. One of the topics in this section is finding the cube or cube root of a number. A cubed number is the solution when.

THE LINEAR FACTORIZATION THEOREM. What is the Linear Factorization Theorem? If where n > 1 and a n ≠ 0 then Where c 1, c 2, … c n are complex numbers.

Do Now Given the polynomial 8 -17x x - 20x 4 – Write the polynomial in standard form: _______________________________________ – Classify the polynomial.

The Fundamental Theorem of Algebra It’s in Sec. 2.6a!!! Homework: p odd, all.

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.

5.2 Polynomials, Linear Factors, and Zeros

Quadratic Formula. Solve x 2 + 3x – 4 = 0 This quadratic happens to factor: x 2 + 3x – 4 = (x + 4)(x – 1) = 0 This quadratic happens to factor: x 2.

5.2 Polynomial & Linear Factors Learning goals graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using.

Solving Polynomials.

Solving Equations Containing First, we will look at solving these problems algebraically. Here is an example that we will do together using two different.

Solving Quadratic Equation by Graphing Students will be able to graph quadratic functions.

Warm up… You’ll need to pick up the worksheet up front. Remember how we used the calculator on Friday. Need to graph the vertex along with four other.

Topic VII: Polynomial Functions 7.3 Solving Polynomial Equations.

Chapter 2 – Polynomial and Rational Functions 2.2 – Polynomial Functions of Higher Degree.

Holt McDougal Algebra Fundamental Theorem of Algebra Intelligence is knowing that a tomato is a fruit; Wisdom is not putting it in a fruit salad.

Copyright © Cengage Learning. All rights reserved. 4 Complex Numbers.

Chapter 2 – Polynomial and Rational Functions 2.5 – The Fundamental Theorem of Algebra.

Lesson 7.4 Solving polynomial equations in factored form

6.2 Polynomials and Linear Functions

Algebra II Explorations Review ( )

Do Now Given the polynomial 8 -17x3+ 16x - 20x4 - 8

Mod. 3 Day 4 Graphs of Functions.

Sec 4: Factored Polynomials and their Roots

Creating Polynomials Given the Zeros.

Section 4.5: Solving Polynomial Equations

Multiply Fractions Name: _________________________________

Bell Work: 11/6/14 Simplify the following numeric fractions

Polynomials, Linear Factors, Zeros

Polynomial Multiplicity

Solving Polynomial Equations

7.4 Solving factored polynomials

ALGEBRA II ALGEBRA II HONORS/GIFTED - SECTION 5-2 (Polynomials, Linear Functions, and Zeros) 2/17/2019 ALGEBRA II SECTION.

Warm-up: Find all real solutions of the equation X4 – 3x2 + 2 = 0

Rational Root Theorem.

Solving Polynomial Equations in Factored Form

Objective SWBAT solve polynomial equations in factored form.

Presentation transcript:

6.2 Polynomials and Linear Functions

Polynomials and Real Roots Relative Maximum POLYNIOMIAL EQUIVALENTS Roots Zeros Solutions X-Intercepts Relative Maximum Relative Minimum Relative Minimum ROOTS !

Write a polynomial given the roots 0, -3, 3 Put in factored form y = (x – 0)(x + 3)(x – 3) y = (x)(x + 3)(x – 3) y = x(x² – 9) y = x³ – 9x

Write a polynomial given the roots 2, -4, ½ Note that the ½ term becomes (x-1/2). We don’t like fractions, so multiply both terms by 2 to get (2x-1) Put in factored form y = (x – 2)(x + 4)(2x – 1) y = (x² + 4x – 2x – 8)(2x – 1) y = (x² + 2x – 8)(2x – 1) y = 2x³ – x² + 4x² – 2x – 16x + 8 y = 2x³ + 3x² – 18x + 8

Write the polynomial in factored form. Then find the roots Write the polynomial in factored form. Then find the roots. Y = 3x³ – 27x² + 24x Y = 3x³ – 27x² + 24x Y = 3x(x² – 9x + 8) Y = 3x(x – 8)(x – 1) ROOTS? 3x(x – 8)(x – 1) = 0 Roots = 0, 8, 1 8 -8 -1 -9 FACTORED FORM

Write the polynomial in factored form. Then find the roots Write the polynomial in factored form. Then find the roots. Y = x4 + 6x² + 8 8 Y = x4 + 6x² + 8 Y =(x² + 4)(x² + 2) ROOTS? (x² + 4)(x² + 2) = 0 Set each group equal to zero and solve x² + 4 = 0 x² + 2 = 0 x² = -4 x² = -2 X = √-4 X = √-2 X = ±2i x = ±i√2 4 2 6 FACTORED FORM

What is Multiplicity? Multiplicity is when you have multiple roots that are exactly the same. We say that the multiplicity is how many duplicate roots that exist. Ex: (x-2)(x-2)(x+3) Ex: (x-1)4 (x+3) Ex: y =x(x-1)(x+3) Note: two answers are x=2; therefore the multiplicity is 2 Note: four answers are x=1; therefore the multiplicity is 4 Note: there are no repeat roots, so we say that there is no multiplicity

Let’s Try One Find any multiple zeros of f(x)=x4+6x3+8x2 and state the multiplicity

Let’s Try One Find any multiple zeros of f(x)=x4+6x3+8x2 and state the multiplicity