Chapter 7.4 The Remainder and Factor Theorems Standard & Honors

Slides:

Advertisements

Similar presentations

Remainder and Factor Theorems

Advertisements

The Remainder and Factor Theorems Check for Understanding 2.3 – Factor polynomials using a variety of methods including the factor theorem, synthetic division,

Warm Up Divide using long division ÷ ÷

Factoring Polynomials

Warm Up Factor each expression. 1. 3x – 6y 2. a 2 – b 2 3. (x – 1)(x + 3) 4. (a + 1)(a 2 + 1) x 2 + 2x – 3 3(x – 2y) (a + b)(a – b) a 3 + a 2 + a + 1 Find.

Daily Warm-UP Quiz Factor completely: 1. x 4 – 17x x x x 4 – 7x x 8 – x 6 – x 6 – 36.

Dividing Polynomials 6-3 Warm Up Lesson Presentation Lesson Quiz

2.5 Apply the Remainder and Factor Theorems p. 120 How do you divide polynomials? What is the remainder theorem? What is the difference between synthetic.

6.5 The Remainder and Factor Theorems p. 352 How do you divide polynomials? What is the remainder theorem? What is the difference between synthetic substitution.

1 What we will learn today…  How to divide polynomials and relate the result to the remainder and factor theorems  How to use polynomial division.

Section 3-3 Dividing Polynomials Objectives: Use Long Division and Synthetic Division to divide polynomials.

Algebraic long division Divide 2x³ + 3x² - x + 1 by x + 2 x + 2 is the divisor The quotient will be here. 2x³ + 3x² - x + 1 is the dividend.

The Remainder and Factor Theorems

7.4 THE REMAINDER & FACTOR THEOREMS Objectives: The student will be able to… 1)evaluate functions using synthetic substitution 2)determine whether a binomial.

UNIT 2 – QUADRATIC, POLYNOMIAL, AND RADICAL EQUATIONS AND INEQUALITIES Chapter 6 – Polynomial Functions 6.7 – The Remainder and Factor Theorems.

The Remainder and Factor Theorems 6.5 p When you divide a Polynomial f(x) by a divisor d(x), you get a quotient polynomial q(x) with a remainder.

6-7 The Division Algorithm & The Remainder Theorem dividend=quotient. divisor + remainder If a polynomial f(x) is divided by x - c, the remainder is the.

Section 5.3(d) Synthetic Substitution. Long division Synthetic Division can be used to find the value of a function. This process is called Synthetic.

Holt McDougal Algebra 2 Dividing Polynomials How do we use long division and synthetic division to divide polynomials?

Division of Polynomials Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Dividing Polynomials Long division of polynomials.

6.5 Warm Up 1.Factor 8x Factor 5x x 2 – x – 2. 3.Factor 200x 6 – 2x 4. 4.Find the product of (2x – 3)(2x – 5). 5.Find the product of (5x.

Warm Up Divide using long division, if there is a remainder put it as a fraction ÷ ÷ x + 5y 23 7a – b Divide. 6x – 15y 3 7a 2.

Chapter Dividing polynomials. Objectives  Use long division and synthetic division to divide polynomials.

Then/Now You used the Distributive Property and factoring to simplify algebraic expressions. Evaluate functions by using synthetic substitution. Determine.

Polynomial & Synthetic Division Algebra III, Sec. 2.3 Objective Use long division and synthetic division to divide polynomials by other polynomials.

Holt Algebra Dividing Polynomials 6-3 Dividing Polynomials Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson.

Dividing Polynomials 6-3 Warm Up Lesson Presentation Lesson Quiz

Dividing Polynomials 6-3 Warm Up Lesson Presentation Lesson Quiz

Dividing Polynomials 3-3 Warm Up Lesson Presentation Lesson Quiz

Warm-ups Week 8 10/8/12 Find the zeros of f(x) = x3 + 2x2 – 13x + 10 algebraically (without a graphing calculator). What if I told.

Polynomial Long Division Review

Factoring Polynomials

Factoring Polynomials

Factoring Polynomials

Polynomial Long Division Review

Polynomial Long Division Review

Factoring Polynomials

Remainder and Factor Theorems

The Remainder and Factor Theorems

7.4 The Remainder and Factor Theorems

Dividing Polynomials Warm Up Lesson Presentation Lesson Quiz

DIVIDING POLYNOMIALS Synthetically!

Chapter 7.6 Rational Zero Theorem Standard & Honors

Do Now Graph the following using a calculator: A) B)

Factoring Polynomials

Warm Up 1. Divide f(a) = 4a2 – 3a + 6 by a – 2 using any method.

Polynomial Division; The Remainder Theorem and Factor Theorem

Factoring Polynomials

Chapter 7.5 Roots and Zeros Standard & Honors

Chapter 6.4 Completing the Square Standard & Honors

Chapter 6.3 Solving Quadratic Functions by Factoring Standard & Honors

Warm-up: Divide using Long Division

Remainder and Factor Theorem

Review Chapter 7.7, 7.8 &11.7 Standard & Honors

Dividing Polynomials 6-3 Warm Up Lesson Presentation Lesson Quiz

Solving Equations using Quadratic Techniques

Factoring Polynomials

The Remainder and Factor Theorems

You can use synthetic division to evaluate polynomials

Dividing Polynomials 6-3 Warm Up Lesson Presentation Lesson Quiz

Factoring Polynomials

The Remainder and Factor Theorems

4-3: Remainder and Factor Theorems

21 = P(x) = Q(x) . (x - r) + P(r) 4.3 The Remainder Theorem

Dividing Polynomials 6-3 Warm Up Lesson Presentation Lesson Quiz

5.5 Apply the Remainder and Factor Theorems

Factoring Polynomials

Presentation transcript:

Chapter 7.4 The Remainder and Factor Theorems Standard & Honors Algebra II Mr. Gilbert Chapter 7.4 The Remainder and Factor Theorems Standard & Honors 11/3/2018

Agenda Warm up Home Work Lesson Practice Homework 11/3/2018

Homework Review 11/3/2018

Communicate Effectively Synthetic substitution: When synthetic division is used to evaluate a function. Depressed polynomial: The quotient when a polynomial is divided by one of its binomial factors. For example: divide x4+x3-17x2-20x+32 by x-4 the depressed polynomial is x3+5x2+3x-8 The Factor Theorem: The binomial x-a is a factor of f(x) iff f(a)=0. The Remainder Theorem: If a polynomial f(x) is divided by x-a, the remainder is the constant f(a) where q(x) is a polynomial with degree one less than f(x). 11/3/2018

Example 1 Synthetic Substitution (3) Example 2 Use the Factor Theorem (3) Example 3 Find All Factors of a Polynomial (3) 11/3/2018 Lesson 4 Contents

3 10 41 164 654 If find f (4). Method 1 Synthetic Substitution By the Remainder Theorem, f (4) should be the remainder when you divide the polynomial by x – 4. Notice that there is no x term. A zero is placed in this position as a placeholder. 3 10 41 164 654 Answer: The remainder is 654. Thus, by using synthetic substitution, f (4) = 654. 11/3/2018 Example 4-1a

Method 2 Direct Substitution Replace x with 4. Original function Replace x with 4. Simplify. or 654 Answer: By using direct substitution, f (4) = 654. 11/3/2018 Example 4-1b

If find f (3). Answer: 34 11/3/2018 Example 4-1c

Show that is a factor of Then find the remaining factors of the polynomial. The binomial x – 3 is a factor of the polynomial if 3 is a zero of the related polynomial function. Use the factor theorem and synthetic division. 1 7 6 0 11/3/2018 Example 4-2a

Since the remainder is 0, (x – 3) is a factor of the polynomial Since the remainder is 0, (x – 3) is a factor of the polynomial. The polynomial can be factored as The polynomial is the depressed polynomial. Check to see if this polynomial can be factored. Factor the trinomial. Answer: So, 11/3/2018 Example 4-2b

Check. You can see that the graph of the related function Check You can see that the graph of the related function crosses the x-axis at 3, –6, and –1. Thus, 11/3/2018 Example 4-2c

Show that is a factor of Then find the remaining factors of the polynomial. Answer: 1 6 5 0 So, Since 11/3/2018 Example 4-2d

Geometry The volume of a rectangular prism is given by Find the missing measures. The volume of a rectangular prism is You know that one measure is x – 2, so x – 2 is a factor of V(x). 1 9 20 0 11/3/2018 Example 4-3a

The quotient is . Use this to factor V(x). Volume function Factor. Factor the trinomial Answer: The missing measures of the prism are x + 4 and x + 5. 11/3/2018 Example 4-3b

Answer: The missing measures of the prism are x – 2 and x + 5. Geometry The volume of a rectangular prism is given by Find the missing measures. Answer: The missing measures of the prism are x – 2 and x + 5. 11/3/2018 Example 4-3c

Homework See Syllabus 7.4 11/3/2018