Find the roots Identify the multiplicity 3.5: Finding Real Roots of Polynomial Equations.

Slides:

Advertisements

Similar presentations

Finding Real Roots of Polynomial Equations 6-5

Advertisements

6.5 & 6.6 Theorems About Roots and the Fundamental Theorem of Algebra

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.

16 Days. Two Days  Review - Use FOIL and the Distributive Property to multiply polynomials.

EXAMPLE 2 Find all real zeros of f (x) = x 3 – 8x 2 +11x SOLUTION List the possible rational zeros. The leading coefficient is 1 and the constant.

Section 3-5 Finding Real Roots of Polynomial Equations

Station A: Linear Functions: Solve each equation Solve for the specified variable: Write an equation in slope-intercept form for the line.

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

Finding Real Roots of Polynomial Equations 6-5

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.

GUIDED PRACTICE for Example How many solutions does the equation

Real Zeros of a Polynomial Function Objectives: Solve Polynomial Equations. Apply Descartes Rule Find a polynomial Equation given the zeros.

7.5.1 Zeros of Polynomial Functions

The Rational Root Theorem The Rational Root Theorem gives us a tool to predict the Values of Rational Roots:

Finding Real Roots of Polynomial Equations

Objectives Identify the multiplicity of roots.

Objectives Identify the multiplicity of roots Use the Rational Root Theorem and the Irrational Root Theorem to solve polynomial equations.

7.4 and 7.5 Solving and Zeros of Polynomials

Solving Quadratic Equations – Part 1 Methods for solving quadratic equations : 1. Taking the square root of both sides ( simple equations ) 2. Factoring.

Zeros of Polynomials 2.5.

Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.

Holt Algebra Solving Quadratic Equations by Graphing and Factoring A trinomial (an expression with 3 terms) in standard form (ax 2 +bx + c) can be.

7.4 Solving Polynomial Equations

Theorems About Roots of Polynomial Equations. Find all zeros: f(x)= x +x –x Synthetic Division one zero…need 2 more use (x – k), where.

EXAMPLE 5 Use the result to write f (x) as a product of two factors. Then factor completely. f (x) = x 3 – 2x 2 – 23x + 60 The zeros are 3, – 5, and 4.

Objectives: 1. Use the factor theorem. 2. Factor a polynomial completely.

Finding Rational Zeros (Day 1)

Solving Polynomials. What does it mean to solve an equation?

Essential Questions How do we identify the multiplicity of roots?

Finding Real Roots of Polynomial Equations 3-5

7.6 Rational Zero Theorem Depressed equation. All the possible rational Zeros To find all the possible rational zero, take all the factors of the last.

LESSON 5.6 Rational Zeros of Polynomial Functions.

©thevisualclassroom.com To solve equations of degree 2, we can use factoring or use the quadratic formula. For equations of higher degree, we can use the.

EXAMPLE 3 Find zeros when the leading coefficient is not 1

Solving polynomial equations

Slide Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc.

Ch. 6.4 Solving Polynomial Equations. Sum and Difference of Cubes.

GRAPHING LINEAR AND NONLINEAR SYSTEMS A-CED.2 CREATE EQUATIONS IN TWO OR MORE VARIABLES TO REPRESENT RELATIONSHIPS BETWEEN QUANTITIES; GRAPH EQUATIONS.

Warm-Up Solve each equation by factoring. 1) x x + 36 = 02) 2x 2 + 5x = 12.

Solving Quadratic Functions by Factoring. Factoring Review.

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.

Algebra Finding Real Roots of Polynomial Equations.

HOMEWORK CHECK.

Warm Ups Term 2 Week 6.

Warm Up Factor completely. 1. 2y3 + 4y2 – 30y 2y(y – 3)(y + 5)

Solving Polynomial Equations

8-6 Solving Quadratic Equations using Factoring

Let’s Review the Quadratic Formula & Completing the Square!!

Real Zeros Intro - Chapter 4.2.

Finding Real Roots of Polynomial Equations

Finding polynomial roots

9.6 Perfect Squares & Factoring

Creating Polynomials Given the Zeros.

SECTION 9-3 : SOLVING QUADRATIC EQUATIONS

Have out to be checked: P. 680/14-23 all, 29; Don't graph 22 and 23.

1. Use the quadratic formula to find all real zeros of the second-degree polynomial

7.4 and 7.5 Solving and Zeros of Polynomials

You can find the roots of some quadratic equations by factoring and applying the Zero Product Property. Functions have zeros or x-intercepts. Equations.

Warm Up Factor completely. 1. 2y3 + 4y2 – 30y 2y(y – 3)(y + 5)

(a) long division (b)synthetic division

WARM – UP 1. Find all of the real roots of the function:

Warm Up Use long division to find (6

2.6 Find Rational Zeros Pg. 89.

Solving Special Cases.

2.6 Find Rational Zeros Pg. 89.

1) Find f(g(x)) and g(f(x) to show that f(x) and g(x) are inverses

Preview to 6.7: Graphs of Polynomial

Presentation transcript:

Find the roots Identify the multiplicity 3.5: Finding Real Roots of Polynomial Equations

Common Core Standards: CC.9-12.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. CC.9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. CC.9-12.A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, … Include cases where f(x) and/or g(x) are linear, polynomial, rational, … functions. 3.5: Finding Real Roots of Polynomial Equations

Solving a Polynomial: P(x) = 0 If (x-a) is a factor of P(x), then x=a is a solution. If x=a is a solution of P(x), then (x-a) is a factor. (Factor Theorem)

Example

Steps to Solve a Polynomial Equation: 1.Factor out any GCF 2.Graph and identify a root. 3.Use synthetic division to prove it is a root and simplify the equation down to a quadratic. 4.Finish solving by using the quadratic formula (or factoring). 5.CHECK your answers (look at the graph!)

Solve the polynomial equation by factoring. Example 1: Using Factoring to Solve Polynomial Equations 4x 6 + 4x 5 – 24x 4 = 0 Factor out the GCF, 4x 4. Factor the quadratic or use Quadratic Formula. Set each factor equal to 0. Solve for x. The roots are 0, –3, and 2.

Example 1 Continued Check Use a graph. The roots appear to be located at x = 0, x = –3, and x = 2.

Solve the polynomial equation by factoring. Example 1: Using Factoring to Solve Polynomial Equations x = 26x 2 Set each factor equal to 0. Solve for x. The roots are 5, –5, 1, and -1. Set the equation equal to 0. Factor the trinomial. Factor - difference of squares.

Multiplicity

3x x x 3 = 0 Factored form:3x 3 (x + 3) 2 =0 x is a factor three times. Therefore, the multiplicity of the root (0) is… (x+3) is a factor two times. Therefore, the multiplicity of the root (-3) is… Multiplicity

You cannot always determine the multiplicity of a root from a graph. You determine multiplicity when the polynomial is in factored form. 3x x x 3 = 0 3x 3 (x + 3) 2 =0

Identify the roots of each equation. State the multiplicity of any root. 4x 6 + 4x 5 – 24x 4 = 0 4x 4 (x + 3)(x – 2) = 0 The roots are 0, –3, and 2. Zero has a multiplicity of 4. 4x 4 (x 2 + x – 6) = 0

Identify the roots of each equation. State the multiplicity of any root. x 3 + 3x x = 0 x (x + 5)(x – 2) = 0 The roots are 0, –5, and 2. No Multiplicities in these zeros.

Identify the roots of each equation. State the multiplicity of any root. (x – 1)(x + 3)(x + 3)(x + 3)=0 The roots are 1 and -3. The root –3 has a multiplicity of 3. x 4 + 8x x 2 – 27 = 0

Solving a Polynomial: P(x) = 0 If (x-a) is a factor of P(x), then x=a is a solution. If x=a is a solution of P(x), then (x-a) is a factor. (Factor Theorem)

Given the roots of a polynomial, find the polynomial:

Homework 3.5, p 186: # WS - Odds

HA2 Homework 3.5, pp : # 15-26, 44