3.3 SPECIAL Factoring 12/5/2012. Perfect Squares 11 1 42 2 93 3 164 4 255 5 366 6 49 7 7 648 8 819 9 100 10 121 11 144 12 169 13.

Slides:

Advertisements

Similar presentations

Chapter 6 Section 4: Factoring and Solving Polynomials Equations

Advertisements

Bell Problem Perform the indicated operation. (x -5)(x2 – 5x + 7)

Multiplying a binomial by a monomial uses the Distribute property Distribute the 5.

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

Factoring Sums & Differences of Cubes

6.5 Factoring Cubic Polynomials 1/31/2014. Cube: a geometric figure where all sides are equal. 10 in Volume of a cube: side sideside V= V = 1000.

6.5 Factoring Cubic Polynomials

6.4 Factoring Polynomial Equations * OBJ: Factor sum & difference of cubes Do Now: Factor 1) 25x 2 – 492) x 2 + 8x + 16 (5x + 7)(5x – 7) (x + 4)(x + 4)

Factor Special Products April 4, 2014 Pages

Warm Up #10 Multiply the polynomial. 1. (x + 2)(x + 3)(x + 1)

Perfect Square Trinomials and Difference of Perfect Squares

Review: Factoring. Factors Remember that “factors” of a number are the numbers that can be multiplied to get that number. Factors of 20 are 1 and 20,

HW: 6.2 Practice Worksheet. EXAMPLE 1 Add polynomials vertically and horizontally a. Add 2x 3 – 5x 2 + 3x – 9 and x 3 + 6x in a vertical format.

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.

Objective: 6.4 Factoring and Solving Polynomial Equations 1 5 Minute Check  Simplify the expression

Factoring and Solving Polynomial Equations (Day 1)

EXAMPLE 3 Multiply polynomials vertically and horizontally a. Multiply – 2y 2 + 3y – 6 and y – 2 in a vertical format. b. Multiply x + 3 and 3x 2 – 2x.

Day Problems Simplify each product. 1. 8m(m + 6) 2. -2x(6x3 – x2 + 5x)

Factoring - Difference of Squares What is a Perfect Square.

factoring special products Formulas to memorize!

Factoring Special Products. Factoring: The reverse of multiplication Use the distributive property to turn the product back into factors. To do this,

2.3Special Products of Polynomials Square of a Binomial Pattern Multiply binomials by using F O I L.

Factoring Polynomials: Part 1 GREATEST COMMON FACTOR (GCF) is the product of all prime factors that are shared by all terms and the smallest exponent of.

Warm-Up #2 Multiply these polynomials. 1) (x-5) 2 2) (8x-1) 2 3. (4x- 3y)(3x +4y) Homework: P5 (1,3,5,11,13,17,27,33,41, 45,49,55,59,63,71,73,77) Answers:

GOAL: FACTOR CUBIC POLYNOMIALS AND SOLVE CUBIC EQUATIONS Section 6-5: Factoring Cubic Polynomials.

GCF present Distributive Property in Reverse.

Warm - up Factor: 1. 4x 2 – 24x4x(x – 6) 2. 2x x – 21 (2x – 3)(x + 7) 3. 4x 2 – 36x + 81 (2x – 9) 2 Solve: 4. x x + 25 = 0x = x 2 +

Section 6.3 Special Factoring. Overview In this section we discuss factoring of special polynomials. Special polynomials have a certain number of terms.

Copyright © Cengage Learning. All rights reserved. Factoring Polynomials and Solving Equations by Factoring 5.

Strategies for Factoring

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.

EXAMPLE 1 Find a common monomial factor Factor the polynomial completely. a. x 3 + 2x 2 – 15x Factor common monomial. = x(x + 5)(x – 3 ) Factor trinomial.

Warm Up:. Factoring Polynomials Number of TermsFactoring TechniqueGeneral Pattern Any number of terms Greatest Common Factora 3 b 2 + 2ab 2 = ab 2 (a.

7.6 Polynomials and Factoring Part 2: Factoring. Factoring The process of finding polynomials whose product equals a given polynomial is called factoring.

5.5 Factoring Special Patterns 12/12/12. Perfect Squares

Warm Up Day 2 Unit 3A. Use FOIL to Simplify: 1. (x-2)(x+2) 2. (5x + 6)(5x – 6) 3. (x -3)(x+3) 4. (x-1)(x+1)

Binomial X Binomial The problems will look like this: (x – 4)(x + 9)

Factor and Solve Polynomial Equations Homework Questions?

Special Cases of Factoring. 1. Check to see if there is a GCF. 2. Write each term as a square. 3. Write those values that are squared as the product of.

1 Copyright © Cengage Learning. All rights reserved.

6 – 3 Adding, Subtracting and Multiplying Polynomials Day 1 Objective: Add, subtract, and multiply polynomials.

Notes Over 6.3 Adding Polynomial Horizontally and Vertically Find the sum. Just combine like terms.

EXAMPLE 3 Multiply polynomials vertically and horizontally a. Multiply –2y 2 + 3y – 6 and y – 2 in a vertical format. b. Multiply x + 3 and 3x 2 – 2x +

5.3C- Special Patterns for Multiplying Binomials SUM AND DIFFERENCE (a+b)(a-b) = a² - b² (x +2)(x – 2) = x² -4 “O & I” cancel out of FOIL SQUARE OF A BINOMIAL.

Module 3.3 Factoring.

Section 6.4: Factoring Polynomials

Copyright © 2013, 2009, 2006 Pearson Education, Inc.

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

Special Products of Polynomials

8-4 Special Products of Polynomials

Warm Up Subtract: Add:.

Warm - up x2 – 24x 4x(x – 6) 2. 2x2 + 11x – 21 (2x – 3)(x + 7)

Factoring Polynomial Functions

8.6 Choosing a Factoring Method

Multiply together.

Factoring Sums & Differences of Cubes

Chapter 6 Section 4.

Special Factoring Formulas & a General Review of Factoring

Algebra 1 Section 10.1.

Unit 5 Factor Special Products

5.4 Factor and Solve Polynomial Equations

Example 2A: Factoring by GCF and Recognizing Patterns

5.5: Factoring the Sum and Difference of Two Cubes

(B12) Multiplying Polynomials

2.3 Factor and Solve Polynomial Expressions Review (cont.)

6.6 Factoring Polynomials

Factoring Cubes and Factoring by Grouping

Factoring Polynomials First: Look for a GCF 4 Second: Number of Terms 2 3 Cubes Squares Perfect Square Trinomial Grouping X 2 – 9 X 3 – 27 = (x - 3)

Do Now 3/4/19 Take out your HW from last night.

Factoring Polynomials, Special Cases

Presentation transcript:

3.3 SPECIAL Factoring 12/5/2012

Perfect Squares

Review Find the product using FOIL 1.(x + 2) (x – 2) Answer: x 2 – 4 2. (x + 5) (x – 5) Answer: x 2 – (2x – 3) (2x + 3) Answer: 4x 2 – 9 What’s the pattern???

Difference of Two Squares Pattern (a + b) (a – b) = a 2 – b 2 In reverse, a 2 – b 2 gives you (a + b) (a – b) Examples: 1. x 2 – 4 = x 2 – 2 2 = (x + 2) (x – 2) 2. x 2 – 144 =(x + 12) (x – 12) 3. 4x 2 – 25 = (2x + 5) (2x – 5)

1000 = = = = = 6 3 Perfect Cubes 125 = = = = = 1 3

The sum of two cubes: The difference of two cubes:

Factor the Sum or Difference of Two Cubes a.Factor. x 3x b.Factor. 8p 38p 3 – q 3q 3 SOLUTION Write as sum of two cubes. x 3x = x 3x a. () 4x + () x 2x 2 4x4x+ – 4242 = Use special product pattern. () 4x + () x 2x 2 4x4x+ – 16 = Simplify.

Factor the Sum or Difference of Two Cubes = – () q2p2p + q2q2 2pq+ 4p24p2 () Simplify. b. 8p 38p 3 – q 3q 3 – () 2p 3 q 3q 3 = Write as difference of two cubes. = – () q2p2p + q2q2 2pq2pq [] () 2p 2 + Use special product pattern.

Checkpoint Factor the polynomial. 1. x 3x x 3 +8 ANSWER () 1x + () x 2x 2 x+ – 1 () 25x5x + () 25x 2 10x+ – 4 3. x 3x – () 6x + () x 2x 2 6x6x+36 –

Factor Polynomials with GCF a. Factor 16x 4 2x.2x. – Take out GCF. = () 2x2x 8x 38x 3 1 – a. 16x 4 2x2x – Use a 3 –b 3 pattern. = () 2x2x 2x2x 1 – 4x 24x 2 2x2x1 ++ ()

Factor by Grouping Factor the polynomial. b.a. x 2x 2 () 1x – () 1x – 9 – x 3x 3 2x 22x 2 16x –– 32 + SOLUTION Use distributive property. a. x 2x 2 () 1x – () 1x – 9 – = () 9x 2x 2 – () 1x – Difference of two squares = () 3x – () 3x+ () 1x –

Factor by Grouping Factor each group. = ) x 2x 2 – ( – ( x )) – 2 ( x Use distributive property. = ) – 16 () – 2 ( xx 2x 2 Difference of two squares = () 4x – () 4x+ () 2x – Group terms. b. = () x 3x 3 – () 32 – x 3x 3 2x 22x 2 16x –– x 22x x +

Checkpoint Factor the polynomial by grouping. 8. Factor by Grouping x 2x 2 () 6x+ () 6x+4 – 9. x 3x 3 4x 24x 2 25x –– x 3x 3 3x 23x 2 4x4x ANSWERS () 2x – () 2x+ () 6x+ () 5x – () 5x+ () 4x – () 3x+ () 4x 2x 2 +

Homework: Worksheet 3.3