X 4 – 81 = (x 2 + 9) (x + 3) (x-3) Solving Polynomial Equations At the end of this two-part lesson, you will:  Solve Polynomial Equations by Factoring/Using.

Slides:

Advertisements

Similar presentations

solution If a quadratic equation is in the form ax 2 + c = 0, no bx term, then it is easier to solve the equation by finding the square roots. Solve.

Advertisements

Solving Quadratic Equations Algebraically Lesson 2.2.

If b2 = a, then b is a square root of a.

Lesson 1-6 Solving Quadratic Equations. Objective:

2-4 completing the square

Solving Quadratic Equations by Completing the Square

Solving Quadratics by Completing the Square, continued Holt Chapter 5 Section 4.

Section 10.5 – Page 506 Objectives Use the quadratic formula to find solutions to quadratic equations. Use the quadratic formula to find the zeros of a.

6.2 Polynomials and Linear Factors

Quadratic Inequalities Lesson Definition Recall the quadratic equation ax 2 + bx + c = 0 Replace = sign with, ≤, or ≥ makes it a quadratic inequality.

Chapter 7 Quadratic Equations and Functions

Solving Quadratic Functions Lesson 5.5b. Finding Zeros Often with quadratic functions f(x) = a*x 2 + bx + c we speak of “finding the zeros” This means.

Algebra 1B Chapter 9 Solving Quadratic Equations By Graphing.

CA STANDARDS 20.0: Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations. Agenda 1.) Lesson.

The Quadratic Formula. What does the Quadratic Formula Do ? The Quadratic formula allows you to find the roots of a quadratic equation (if they exist)

Unit 2 – Quadratic, Polynomial, and Radical Equations and Inequalities Chapter 5 – Quadratic Functions and Inequalities 5.2 – Solving Quadratic Equations.

Section 4.7 – The Quadratic Formula Students will be able to: To solve equations using the Quadratic Formula To determine the number of solutions by using.

What you will learn How to solve a quadratic equation using the quadratic formula How to classify the solutions of a quadratic equation based on the.

Slide Copyright © 2012 Pearson Education, Inc.

WARM UP WHAT TO EXPECT FOR THE REST OF THE YEAR 4 May The Discriminant May 29 Chapter Review May 30 Review May 31 Chapter 9 Test June Adding.

The Quadratic Formula Students will be able to solve quadratic equations by using the quadratic formula.

Graphs of Quadratic Functions Graph the function. Compare the graph with the graph of Example 1.

4.6 Completing the Square Completing a perfect square trinomial allows you to factor the completed trinomial as the square of a binomial. You can solve.

Section 6.4 Solving Polynomial Equations Obj: to solve polynomial equations.

Solving Quadratic Equations by Factoring. The quadratic equation is written in the form ax 2 + bx + c = 0 To solve quadratic equations by factoring we.

Quadratic Equations: Factoring, Square Root Methods.

LESSON 5.6 Rational Zeros of Polynomial Functions.

Algebra 2 Ch.6 Notes Page 43 P Solving Polynomial Equations.

Chapter 4: Polynomial and Rational Functions. 4-2 Quadratic Equations For a quadratic equation in the form ax 2 + bx + c = 0 The quadratic Formula is.

Completing the Square SPI Solve quadratic equations and systems, and determine roots of a higher order polynomial.

Warm Up 1.) What is the graph of the function y = -x 2 + 4x + 1?

6-4 Solving Polynomial Equations. Objectives Solving Equations by Graphing Solving Equations by Factoring.

PreCalculus Section 1.6 Solve quadratic equations by: a. Factoring b. Completing the square c. Quadratic formula d. Programmed calculator Any equation.

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

Do Now: Solve x2 – 20x + 51 = 0 3x2 – 5x – 2 = 0 Today’s Agenda

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

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

Bellwork 1)Write in standard form. 2) 3)A student is solving an equation by completing the square. Write the step in the solution that appears just before.

Lesson 6.5: The Quadratic Formula and the Discriminant, pg. 313 Goals: To solve quadratic equations by using the Quadratic Formula. To use the discriminant.

PreCalculus Section 1. 6 Solve quadratic equations by: a. Factoring b

Completing the Square, Quadratic Formula

The Quadratic Formula..

Warm up Factor the expression.

Chapter 4 Quadratic Equations

7.3 Solving Equations Using Quadratic Techniques

The Quadratic Formula..

Quadratic Formula Solving for X Solving for quadratic equations.

Solving Quadratic Functions

Finding polynomial roots

Intro: We already know the standard form of a quadratic equation is:

Solutions, Zeros, and Roots

The Quadratic Formula..

9.3 Solving Quadratic Equations

Solving Quadratic Functions

The Quadratic Formula.

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

13.3 Completing the Square Objective: To complete a square for a quadratic equation and solve by completing the square.

Main Ideas Key Terms Chapter 2 Section 3 Graphing a Polynomial

Solving Equations using Quadratic Techniques

Solving Quadratic Equations

Objective Solve quadratic equations by graphing.

Warm-Up: September 30 / October 1, 2015 Factor each expression

13.3 Completing the Square Objective: To complete a square for a quadratic equation and solve by completing the square.

Homework Check.

13.3 Completing the Square Objective: To complete a square for a quadratic equation and solve by completing the square.

The Quadratic Formula..

The Quadratic Formula..

quadratic formula. If ax2 + bx + c = 0 then

Presentation transcript:

X 4 – 81 = (x 2 + 9) (x + 3) (x-3) Solving Polynomial Equations At the end of this two-part lesson, you will:  Solve Polynomial Equations by Factoring/Using Quadratic Formula  Solve Polynomial Equations by Graphing X 3 – 125 = 0 (x-5) (x 2 +5x + 25) x = 5; x = ?

RECALL: Quadratic Formula Sums and Differences of Cubes u 3 + v 3 = ___________________________ u 3 - v 3 = ___________________________ x = ( u + v) (u 2 – uv + v 2 ) ( u - v) (u 2 + uv + v 2 ) NOTE: Equation must be in standard form ax 2 + bx + c = 0, where a, b, and c are and a > 0.

Review: Factoring Sums and Differences of Cubes Being familiar with perfect cubes will make factoring sums and differences much easier! x 3 – 512 x x 3 – 343 x x 3 – 8 3 = (x – 8) (x 2 + 8x + 64) x 3 – 7 3 = (x – 7) (x 2 + 7x + 49) x = (x + 5) (x 2 - 5x + 25) x = (x + 6) (x 2 - 6x + 36)

SOLVING A POLYNOMIAL EQUATION Solve 8x = 0. Find all complex roots. You will need to use Quadratic Formula. ( ) 3 + __ 3 =

MENTAL MATH: FACTORING HIGHER-DEGREE POLYNOMIALS BY USING A QUADRATIC FORM x 4 – 3x 2 – 10x x x 4 – 8x x 4 – 17x x 4 – 5x 2 + 6x 4 – 16x x 4 – 7x x 4 + 6x Factor completely.

Examples Solving Higher-Degree Equations by Using a Quadratic Form Solve x 4 – 6x = 0. Solve x 4 – 3x = 0. Solve x x2 +10 = 0. Solve x 4 – 4x = 0. Pick your poison!

ExampleSolving Higher-Degree Polynomial Equations by Using Factoring By Grouping 30X X 2 + 3X + 4 = 0 3X 4 + 3X 2 + 6X 2 + 6X = 0 X X 3 + 4X X = 0 Pick your poison!

Solving by Graphing You can also solve a polynomial equation by graphing each side of the equation separately, and finding the x-values (zeros) at the point(s) of intersection. y = x 3 + 3x 2 – x - 3 Step 1: Graph y 1 = x 3 + 3x 2 & y 2 = x + 3x Step 2: Use the intersect feature to find the x-values (zeros) of the points of intersection. The solutions are -3, 1 & 1. …on a final note!

FINAL CHECKS FOR UNDERSTANDING 1.What does it mean for a polynomial with integer coefficients to be completely factored with respect to the integers? 2. Give an example of a second-degree polynomial that cannot be factored with respect to the integers. 3.Factor completely with respect to the integers: 81x 4 – 1 16x 3 – 4 1 – 64x 3 2x 3 – 8x 2 + 3x - 12

Homework Assignment: Pages odd (calculator) all. Column 1 (42-58 all), 61, 62. Reminder: Chapter 6 Exam on _____________________.