Factoring Polynomials

Slides:

Advertisements

Similar presentations

Warm up Use synthetic division to divide (4x3 – 3x2 + 2x + 1)/ (x – 1) (x3 – x2 – 6)/(x + 2)

Advertisements

This is JEOPARDY!!! It would be wise to be attentive, take appropriate notes, and ask questions.

Solving a cubic function by factoring: using the sum or difference of two cubes. By Diane Webb.

5.3 Division of Polynomials. Dividing a Polynomial by a monomial.  Divide each term of the polynomial by the monomial.

4.4 Notes The Rational Root Theorem. 4.4 Notes To solve a polynomial equation, begin by getting the equation in standard form set equal to zero. Then.

6 – 4: Factoring and Solving Polynomial Equations (Day 1)

5.4 Special Factoring Techniques

Factoring Polynomials. 1.Check for GCF 2.Find the GCF of all terms 3.Divide each term by GCF 4.The GCF out front 5.Remainder in parentheses Greatest Common.

Warm Up Factor each expression. 1. 3x – 6y 3(x – 2y) 2. a2 – b2

I am greater than my square. The sum of numerator and denominator is 5 What fraction am I?

Quick Crisp Review Zeros of a polynomial function are where the x-intercepts or solutions when you set the equation equal to zero. Synthetic and long division.

Objectives Use the Factor Theorem to determine factors of a polynomial. Factor the sum and difference of two cubes.

Factoring Polynomials We will learn how to factor cubic functions using factoring patterns. The factoring patterns we will use are the difference of two.

Chapter factoring polynomials. Objectives Use the Factor Theorem to determine factors of a polynomial. Factor the sum and difference of two.

Factoring Polynomials

5.4 Factoring Polynomials

©thevisualclassroom.com Example 1: (4x 3 – 5x – 6) ÷ (2x + 1) 4x 3 + 0x 2 – 5x – 6 2x + 1 2x22x2 4x 3 + 2x 2 –2x 2 – 5x – x – 4x (restriction) –2x 2 –

Chapter 6 Section 2 Multiplication and Division of Rational Expressions 1.

Polynomial Terms and Operations. EXAMPLE 1 Add polynomials vertically and horizontally a. Add 2x 3 – 5x 2 + 3x – 9 and x 3 + 6x in a vertical.

Factoring and the Factor Theorem Hints to determine each type.

Powers and roots. Square each number a) 7 b) 12 c) 20 d) 9 e) 40 a) 49 b) 144 c) 400 d) 81 e) 1600.

Holt McDougal Algebra 2 Factoring Polynomials How do we use the Factor Theorem to determine factors of a polynomial? How do we factor the sum and difference.

 The remainder theorem states that the remainder that you get when you divide a polynomial P(x) by (x – a) is equal to P(a).  The factor theorem is.

Section 2-2 Synthetic Division; The Remainder and Factor Theorems.

3.6 Day 2 Why Synthetic Division? What use is this method, besides the obvious saving of time and paper?

Section 4-3 The Remainder and Factor Theorems. Remainder Theorem Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant,

Warm Up no 0, 3 x = -3. Homework Questions Section 2.2 Synthetic Division; The Remainder and Factor Theorems Objective: To use synthetic division and.

SECTION 3-4 FACTORING POLYNOMIALS Objectives - Use the Factor Theorem to determine factors of a polynomial - Factor the sum and difference of two cubes.

2.5 Apply the Remainder and Factor Theorem Long Division and Synthetic Division Pg. 85.

5.5 – Apply the Remainder and Factor Theorems The Remainder Theorem provides a quick way to find the remainder of a polynomial long division problem.

©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.

Factoring Polynomials. 1.Check for GCF 2.Find the GCF of all terms 3.Divide each term by GCF 4.The GCF out front 5.Remainder in parentheses Greatest Common.

This is JEOPARDY!!! It would be wise to be attentive, take appropriate notes, and ask questions.

Lesson 11-2 Remainder & Factor Theorems Objectives Students will: Use synthetic division and the remainder theorem to find P(r) Determine whether a given.

Sect. 2-2 Synthetic Division; The remainder and Factor theorems Objective: SWBAT use the synthetic division and to apply the remainder and factor theorems.

4-5 Exploring Polynomial Functions Locating Zeros.

1.3 Factoring Polynomials Definition of Factoring Factoring Integers & Monomials Factoring Polynomials.

Math Project Andy Frank Andrew Trealor Enrico Bruschi.

Polynomial Long Division

Remainder and Factor Theorems

Divide x3 + x2 – 10x + 8 by x+4 using long division.

#2.5 Long Division.

Solving Polynomial Equations

Example: Factor the polynomial 21x2 – 41x No GCF Puzzle pieces for 21x2 x, 21x 3x, 7x Puzzle pieces for 10 1, 10 2, 5 We know the signs.

Factoring Sums and Differences

Chapter 7 Factoring.

4.3 The Remainder & Factor Theorems

1a. Divide using long division. (9x3 – 48x2 + 13x + 3) ÷ (x – 5)

Warm Up Factor each expression. 1. 3x – 6y 3(x – 2y) 2. a2 – b2

Chapter 6 Section 3.

Factoring the Sum & Difference of Two Cubes

Essential Questions How do we use the Factor Theorem to determine factors of a polynomial? How do we factor the sum and difference of two cubes.

Polynomials and Polynomial Functions

LEARNING GOALS – LESSON 6.4

7.3 Products and Factors of Polynomials

Chapter 2 notes from powerpoints

Factoring Polynomials

Objectives Use the Factor Theorem to determine factors of a polynomial. Factor the sum and difference of two cubes.

Polynomials and Polynomial Functions

Chapter 6 Section 3.

5.5: Factoring the Sum and Difference of Two Cubes

Objectives Use the Factor Theorem to determine factors of a polynomial. Factor the sum and difference of two cubes.

Factoring the Sum & Difference of Two Cubes

6.7 Dividing a Polynomial by a Polynomial

6. 10a3 + 15a2 – 20a – x2 – 14x t2 – 15 – 2t 9. 25x2 – n2 + 3n – 9 1. y2 – 5y 2. x2 + 16x a2 – a – 4 –18y2 + y3.

Chapter 6 Section 3.

Presentation transcript:

Factoring Polynomials Divide: (x2 – 5x – 9) ÷ (x – 3) (x ¹ 3) 3 1 – 5 – 9 3 3 – 6 1 – 2 – 15 Ans: x – 2 R – 15 Sub x = 3 into: x2 – 5x – 9 = 32 – 5(3) – 9 = – 15

Remainder Theorem In order to determine the remainder when P(x) is divided by (x – k), replace x by k. P(k) = r In order to determine the remainder when replace x by P(x) is divided by (jx – k), P( ) = r

Example 1: Determine the remainder when 2x3 – 5x2 + 2x – 4 is divided by (x + 2). P(x) = 2x3 – 5x2 + 2x – 4 P(–2) = 2(–2)3 – 5(–2)2 + 2(–2) – 4 P(–2) = 2(–8) – 5(4) – 4 – 4 P(–2) = – 16 – 20 – 4 – 4 P(–2) = – 44 The remainder is – 44

Example 2: Determine the remainder when 4x2 + 2x – 3 is divided by (2x – 1). P(x) = 4x2 + 2x – 3 P( ) = 4( )2 + 2( ) – 3 P( ) = 4( ) + 1 – 3 P( ) = 1 + 1 – 3 P( ) = – 1 \ The remainder is –1

Example 1: Factor x3 – 2x2 – 5x + 6 Factor Theorem A polynomial P(x) has (x – k) as factor if and only if P(k) = 0. Example 1: Factor x3 – 2x2 – 5x + 6 P(x) = x3 – 2x2 – 5x + 6 let x = 1 P(1) = (1)3 – 2(1)2 – 5(1) + 6 P(1) = 1 – 2 – 5 + 6 P(1) = 0 \ x – 1 is a factor

To find another factor, divide (x3 – 2x2 – 5x + 6) by (x – 1) 1 – 2 – 5 6 1 1 – 1 – 6 1 – 1 – 6 (x2 – x – 6) = (x – 3)(x + 2) x3 – 2x2 – 5x + 6 = (x – 1)(x – 3)(x + 2)

Example 2: Factor 2x3 – 3x2 – 3x + 2 P(x) = 2x3 – 3x2 – 3x + 2 let x = 1 P(1) = 2(1)3 – 3(1)2 – 3(1) + 2 P(1) = 2 – 3 – 3 + 2 P(1) = – 2 let x = –1 P(–1) = 2(–1)3 – 3(–1)2 – 3(–1) + 2 P(–1) = –2 – 3 + 3 + 2 P(–1) = 0 \ (x + 1) is a factor

Divide (2x3 – 3x2 – 3x + 2) by (x + 1) 2 – 3 – 3 + 2 –1 – 2 5 – 2 now factor this 2 – 5 2 2x2 – 5x + 2 = (2x – 1)(x – 2) \ 2x3 – 3x2 – 3x + 2 = (x + 1)(2x – 1)(x – 2)

Example: Factor the following: Factoring by Grouping Example: Factor the following: 4x3 – 8x2 – 9x + 18 group them in pairs = 4x2(x – 2) – 9(x – 2) common factor = (x – 2)(4x2 – 9) difference of squares = (x – 2)(2x – 3)(2x + 3)

The sum and difference of cubes: In general: (x3 + y3) = (x + y) (x2 – xy + y2) 1. cube root of each term. 2. square the first root. 3. product of the roots (opposite sign) 4. square of the second root. (x3 – y3) = (x – y)(x2 + xy + y2)

Factor the following: (x3 + 8) = (x + 2) (x2 – 2x + 4) (x3 – 64) = (x – 4) (x2 + 4x + 16) (x3 + 27) = (x + 3) (x2 – 3x + 9) (x3 – 125) = (x – 5) (x2 + 5x + 25)