Dividing Polynomials 6-3 Warm Up Lesson Presentation Lesson Quiz

Slides:



Advertisements
Similar presentations
Dividing Polynomials 6-3 Warm Up Lesson Presentation Lesson Quiz
Advertisements

Long and Synthetic Division of Polynomials Section 2-3.
Dividing Polynomials Objectives
Chapter 6: Polynomials and Polynomial Functions Section 6
EXAMPLE 1 Use polynomial long division
Warm up. Lesson 4-3 The Remainder and Factor Theorems Objective: To use the remainder theorem in dividing polynomials.
Warm Up Divide using long division ÷ ÷
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.
HW: Pg #13-61 eoo.
6.8 Synthetic Division. Polynomial Division, Factors, and Remainders In this section, we will look at two methods to divide polynomials: long division.
Polynomial Division, Factors, and Remainders ©2001 by R. Villar All Rights Reserved.
Dividing Polynomials 6-3 Warm Up Lesson Presentation Lesson Quiz
Warm Up Divide using long division ÷ ÷
Dividing Polynomials 6-3 Warm Up Lesson Presentation Lesson Quiz
Section 3-3 Dividing Polynomials Objectives: Use Long Division and Synthetic Division to divide polynomials.
Bell Work Week #16 (12/4/13) Divide using long division. You’ve got 3 minutes ÷ ÷ 12.
Warm up  Divide using polynomial long division:  n 2 – 9n – 22 n+2.
5. Divide 4723 by 5. Long Division: Steps in Dividing Whole Numbers Example: 4716  5 STEPS 1. The dividend is The divisor is 5. Write.
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.
Objective Use long division and synthetic division to divide polynomials.
Ch. 6.3 Dividing Polynomials. Divide x 2 + 2x – 30 by x – 5. ALGEBRA 2 LESSON 6-3 Dividing Polynomials – 30Subtract: (x 2 + 2x) – (x 2 – 5x) = 7x. Bring.
Chapter 1 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Dividing Polynomials; Remainder and Factor Theorems.
6-5: The Remainder and Factor Theorems Objective: Divide polynomials and relate the results to the remainder theorem.
6.3 Dividing Polynomials (Day 1)
12-6 Dividing Polynomials Warm Up Lesson Presentation Lesson Quiz
Objective Use long division and synthetic division to divide polynomials.
6.3 D IVIDING P OLYNOMIAL Use long division and synthetic division to divide polynomials. Use synthetic division to evaluate a polynomial Objective Electricians.
9.4 Polynomial Division, Factors, and Remainders ©2001 by R. Villar All Rights Reserved.
Holt McDougal Algebra 2 Dividing Polynomials How do we use long division and synthetic division to divide polynomials?
Warm Up Divide using long division ÷ Divide.
Holt Algebra Dividing Polynomials Synthetic division is a shorthand method of dividing a polynomial by a linear binomial by using only the coefficients.
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.
Then/Now You factored quadratic expressions to solve equations. (Lesson 0–3) Divide polynomials using long division and synthetic division. Use the Remainder.
Chapter Dividing polynomials. Objectives  Use long division and synthetic division to divide polynomials.
Algebra 2 Divide x 2 + 2x – 30 by x – 5. Lesson 6-3 Dividing Polynomials – 30Subtract: (x 2 + 2x) – (x 2 – 5x) = 7x. Bring down –30. xDivide = x. x – 5.
Holt Algebra Dividing Polynomials 6-3 Dividing Polynomials Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson.
Objective Use long division and synthetic division to divide polynomials.
Divide using long division.
Warm Up Divide using long division ÷ ÷ 2.1 Divide.
Dividing Polynomials 6-3 Warm Up Lesson Presentation Lesson Quiz
Warm Up Divide using long division ÷ Divide.
Dividing Polynomials 3-3 Warm Up Lesson Presentation Lesson Quiz
Warm Up Divide using long division ÷ ÷
Introduction In mathematics, the word “remainder” is often used in relation to the process of long division. You are probably familiar with dividing whole.
Essential Questions How do we use long division and synthetic division to divide polynomials?
Warm Up Divide using long division ÷ ÷
7.4 The Remainder and Factor Theorems
Splash Screen.
Dividing Polynomials Warm Up Lesson Presentation Lesson Quiz
DIVIDING POLYNOMIALS Synthetically!
Do Now Graph the following using a calculator: A) B)
Warm Up 1. Divide f(a) = 4a2 – 3a + 6 by a – 2 using any method.
Apply the Remainder and Factor Theorems Lesson 2.5
Binomial Theorem Honor’s Algebra II.
Dividing Polynomials 3-3 Warm Up Lesson Presentation Lesson Quiz
Objective Use long division and synthetic division to divide polynomials.
Dividing Polynomials 6-3 Warm Up Lesson Presentation Lesson Quiz
Warm-up: Divide using Long Division
Dividing Polynomials 6-3 Warm Up Lesson Presentation Lesson Quiz
Dividing Polynomials 3-3 Warm Up Lesson Presentation Lesson Quiz
Dividing Polynomials 6-3 Warm Up Lesson Presentation Lesson Quiz
You can use synthetic division to evaluate polynomials
Dividing Polynomials 6-3 Warm Up Lesson Presentation Lesson Quiz
Warm up.
Algebra 1 Section 9.6.
Keeper 11 Honors Algebra II
Dividing Polynomials 6-3 Warm Up Lesson Presentation Lesson Quiz
Divide using long division
Warm Up Honors Algebra 2 11/3/17
Presentation transcript:

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

Warm Up Divide using long division. 1. 161 ÷ 7 23 2. 12.18 ÷ 2.1 5.8 6x – 15y 3 3. 2x + 5y 7a2 – ab a 4. 7a – b

Objective “I can…” Use long division and synthetic division to divide polynomials.

Vocabulary synthetic division

Polynomial long division is a method for dividing a polynomial by another polynomials of a lower degree. It is very similar to dividing numbers.

Example 1: Using Long Division to Divide a Polynomial Divide using long division. (–y2 + 2y3 + 25) ÷ (y – 3) Step 1 Write the dividend in standard form, including terms with a coefficient of 0. 2y3 – y2 + 0y + 25 Step 2 Write division in the same way you would when dividing numbers. y – 3 2y3 – y2 + 0y + 25

Example 1 Continued Step 3 Divide. 2y2 + 5y + 15 Notice that y times 2y2 is 2y3. Write 2y2 above 2y3. y – 3 2y3 – y2 + 0y + 25 –(2y3 – 6y2) Multiply y – 3 by 2y2. Then subtract. Bring down the next term. Divide 5y2 by y. 5y2 + 0y –(5y2 – 15y) Multiply y – 3 by 5y. Then subtract. Bring down the next term. Divide 15y by y. 15y + 25 Multiply y – 3 by 15. Then subtract. –(15y – 45) 70 Find the remainder.

Example 1 Continued Step 4 Write the final answer. –y2 + 2y3 + 25 y – 3 = 2y2 + 5y + 15 + 70 y – 3

Check It Out! Example 1a Divide using long division. (15x2 + 8x – 12) ÷ (3x + 1) Step 1 Write the dividend in standard form, including terms with a coefficient of 0. 15x2 + 8x – 12 Step 2 Write division in the same way you would when dividing numbers. 3x + 1 15x2 + 8x – 12

Check It Out! Example 1a Continued Step 3 Divide. 5x + 1 Notice that 3x times 5x is 15x2. Write 5x above 15x2. 3x + 1 15x2 + 8x – 12 –(15x2 + 5x) Multiply 3x + 1 by 5x. Then subtract. Bring down the next term. Divide 3x by 3x. 3x – 12 –(3x + 1) Multiply 3x + 1 by 1. Then subtract. –13 Find the remainder.

Check It Out! Example 1a Continued Step 4 Write the final answer. 15x2 + 8x – 12 3x + 1 = 5x + 1 – 13 3x + 1

Check It Out! Example 1b Divide using long division. (x2 + 5x – 28) ÷ (x – 3) Step 1 Write the dividend in standard form, including terms with a coefficient of 0. x2 + 5x – 28 Step 2 Write division in the same way you would when dividing numbers. x – 3 x2 + 5x – 28

Check It Out! Example 1b Continued Step 3 Divide. x + 8 Notice that x times x is x2. Write x above x2. x – 3 x2 + 5x – 28 –(x2 – 3x) Multiply x – 3 by x. Then subtract. Bring down the next term. Divide 8x by x. 8x – 28 –(8x – 24) Multiply x – 3 by 8. Then subtract. –4 Find the remainder.

Check It Out! Example 1b Continued Step 4 Write the final answer. x2 + 5x – 28 x – 3 = x + 8 – 4 x – 3

Synthetic division is a shorthand method of dividing a polynomial by a linear binomial by using only the coefficients. For synthetic division to work, the polynomial must be written in standard form, using 0 and a coefficient for any missing terms, and the divisor must be in the form (x – a).

Example 2A: Using Synthetic Division to Divide by a Linear Binomial Divide using synthetic division. 1 3 (3x2 + 9x – 2) ÷ (x – ) Step 1 Find a. Then write the coefficients and a in the synthetic division format. 1 3 a = For (x – ), a = . 1 3 1 3 3 9 –2 Write the coefficients of 3x2 + 9x – 2.

Example 2A Continued Step 2 Bring down the first coefficient. Then multiply and add for each column. 1 3 3 9 –2 1 3 Draw a box around the remainder, 1 . 1 3 1 1 3 3 10 Step 3 Write the quotient. 3x + 10 + 1 3 x –

Example 2A Continued 3x + 10 + 1 3 x – Check Multiply (x – ) (x – ) 1 3 3x + 10 + x – = 3x2 + 9x – 2

Example 2B: Using Synthetic Division to Divide by a Linear Binomial Divide using synthetic division. (3x4 – x3 + 5x – 1) ÷ (x + 2) Step 1 Find a. a = –2 For (x + 2), a = –2. Step 2 Write the coefficients and a in the synthetic division format. 3 – 1 0 5 –1 –2 Use 0 for the coefficient of x2.

Example 2B Continued Step 3 Bring down the first coefficient. Then multiply and add for each column. –2 3 –1 0 5 –1 Draw a box around the remainder, 45. –6 14 –28 46 3 –7 14 –23 45 Step 4 Write the quotient. 3x3 – 7x2 + 14x – 23 + 45 x + 2 Write the remainder over the divisor.

Check It Out! Example 2a Divide using synthetic division. (6x2 – 5x – 6) ÷ (x + 3) Step 1 Find a. a = –3 For (x + 3), a = –3. Step 2 Write the coefficients and a in the synthetic division format. –3 6 –5 –6 Write the coefficients of 6x2 – 5x – 6.

Check It Out! Example 2a Continued Step 3 Bring down the first coefficient. Then multiply and add for each column. –3 6 –5 –6 Draw a box around the remainder, 63. –18 69 6 –23 63 Step 4 Write the quotient. 6x – 23 + 63 x + 3 Write the remainder over the divisor.

Check It Out! Example 2b Divide using synthetic division. (x2 – 3x – 18) ÷ (x – 6) Step 1 Find a. a = 6 For (x – 6), a = 6. Step 2 Write the coefficients and a in the synthetic division format. 6 1 –3 –18 Write the coefficients of x2 – 3x – 18.

Check It Out! Example 2b Continued Step 3 Bring down the first coefficient. Then multiply and add for each column. 6 1 –3 –18 There is no remainder. 6 18 1 3 Step 4 Write the quotient. x + 3

You can use synthetic division to evaluate polynomials You can use synthetic division to evaluate polynomials. This process is called synthetic substitution. The process of synthetic substitution is exactly the same as the process of synthetic division, but the final answer is interpreted differently, as described by the Remainder Theorem.

Example 3A: Using Synthetic Substitution Use synthetic substitution to evaluate the polynomial for the given value. P(x) = 2x3 + 5x2 – x + 7 for x = 2. 2 2 5 –1 7 Write the coefficients of the dividend. Use a = 2. 4 18 34 2 9 17 41 P(2) = 41 Check Substitute 2 for x in P(x) = 2x3 + 5x2 – x + 7. P(2) = 2(2)3 + 5(2)2 – (2) + 7 P(2) = 41 

Example 3B: Using Synthetic Substitution Use synthetic substitution to evaluate the polynomial for the given value. 1 3 P(x) = 6x4 – 25x3 – 3x + 5 for x = – . – 1 3 6 –25 0 –3 5 Write the coefficients of the dividend. Use 0 for the coefficient of x2. Use a = . 1 3 –2 9 –3 2 6 –27 9 –6 7 P( ) = 7 1 3

Check It Out! Example 3a Use synthetic substitution to evaluate the polynomial for the given value. P(x) = x3 + 3x2 + 4 for x = –3. –3 1 3 0 4 Write the coefficients of the dividend. Use 0 for the coefficient of x2 Use a = –3. –3 1 4 P(–3) = 4 Check Substitute –3 for x in P(x) = x3 + 3x2 + 4. P(–3) = (–3)3 + 3(–3)2 + 4 P(–3) = 4 

Check It Out! Example 3b Use synthetic substitution to evaluate the polynomial for the given value. 1 5 P(x) = 5x2 + 9x + 3 for x = . 1 5 5 9 3 Write the coefficients of the dividend. Use a = . 1 5 1 2 5 10 5 P( ) = 5 1 5

Example 4: Geometry Application Write an expression that represents the area of the top face of a rectangular prism when the height is x + 2 and the volume of the prism is x3 – x2 – 6x. The volume V is related to the area A and the height h by the equation V = A  h. Rearranging for A gives A = . V h x3 – x2 – 6x x + 2 A(x) = Substitute. –2 1 –1 –6 0 Use synthetic division. The area of the face of the rectangular prism can be represented by A(x)= x2 – 3x. –2 6 1 –3

Check It Out! Example 4 Write an expression for the length of a rectangle with width y – 9 and area y2 – 14y + 45. The area A is related to the width w and the length l by the equation A = l  w. y2 – 14y + 45 y – 9 l(x) = Substitute. 9 1 –14 45 Use synthetic division. 9 –45 1 –5 The length of the rectangle can be represented by l(x)= y – 5.

Lesson Quiz 1. Divide by using long division. (8x3 + 6x2 + 7) ÷ (x + 2) 8x2 – 10x + 20 – 33 x + 2 2. Divide by using synthetic division. (x3 – 3x + 5) ÷ (x + 2) x2 – 2x + 1 + 3 x + 2 3. Use synthetic substitution to evaluate P(x) = x3 + 3x2 – 6 for x = 5 and x = –1. 194; –4 4. Find an expression for the height of a parallelogram whose area is represented by 2x3 – x2 – 20x + 3 and whose base is represented by (x + 3). 2x2 – 7x + 1