6.5 Factoring Cubic Polynomials

Slides:

Advertisements

Similar presentations

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

Advertisements

Chapter 6 Section 4: Factoring and Solving Polynomials Equations

Find a common monomial factor

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

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

Standardized Test Practice

© 2007 by S - Squared, Inc. All Rights Reserved.

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.

Polynomials and Factoring Review By: Ms. Williams.

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

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.

Factoring and Solving Polynomial Equations Chapter 6.4.

Objectives I will use the distributive property to factor a polynomial.

2.4 Factor and Solve Polynomial Equations p. 111 Name two special factoring patterns for cubes. Name three ways to factor a polynomial. What is the difference.

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

Factoring and Solving Polynomial Equations (Day 1)

EXAMPLE 3 Factor by grouping Factor the polynomial x 3 – 3x 2 – 16x + 48 completely. x 3 – 3x 2 – 16x + 48 Factor by grouping. = (x 2 – 16)(x – 3) Distributive.

3.3 SPECIAL Factoring 12/5/2012. Perfect Squares

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

Objective - To recognize and factor a perfect square trinomial. Find the area of the square in terms of x. Perfect Square Trinomial.

5.4 Factor and Solve Polynomial Equations. Find a Common Monomial Factor Monomial: means one term. (ex) x (ex) x 2 (ex) 4x 3 Factor the Polynomial completely.

Warm-Up Exercises Multiply the polynomial. 1.(x + 2)(x + 3) ANSWER x 2 + 5x + 6 ANSWER 4x 2 – 1 2.(2x – 1)(2x + 1) 3. (x – 7) 2 ANSWER x 2 – 14x + 49.

WARM UP Find the product. 1.) (m – 8)(m – 9) 2.) (z + 6)(z – 10) 3.) (y + 20)(y – 20)

5-4 Factoring Quadratic Expressions M11.A.1.2.1: Find the Greatest Common Factor and/or the Least Common Multiple for sets of monomials M11.D.2.1.5: Solve.

WARM UP SOLVE USING THE QUADRATIC EQUATION, WHAT IS THE EXACT ANSWER. DON’T ROUND.

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

Entry Task What is the polynomial function in standard form with the zeros of 0,2,-3 and -1?

Perfect square trinomial x x + 25 = ( x + 5) 2

Solving Quadratic Equations by Factoring Lesson 5.2.

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 +

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

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.

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

Section 5-5: Factoring Using Special Patterns Goal: Factor Using Special Patterns.

Factor and Solve Polynomial Equations Homework Questions?

Factoring Polynomial Expressions Previously, you learned how to factor the following types of quadratic expressions. TypeExample General trinomial Perfect.

EXAMPLE 3 Factor by grouping Factor the polynomial x 3 – 3x 2 – 16x + 48 completely. x 3 – 3x 2 – 16x + 48 Factor by grouping. = (x 2 – 16)(x – 3) Distributive.

Chapter 9 Final Exam Review. Add Polynomials (2x² + x³ – 1) (2x² + x³ – 1) Like Terms terms that have the same variable (2x³ – 5x² + x) + (2x³ – 5x² +

Chapter 5 Section 4. EXAMPLE 1 Find a common monomial factor Factor the polynomial completely. a. x 3 + 2x 2 – 15x Factor common monomial. = x(x + 5)(x.

Objective  SWBAT solve polynomial equations. Section 9.4 “Solve Polynomial Equations in Factored Form” If ab = 0, then a = 0 or b = 0. The zero-product.

Solve Polynomial Equations in Factored Form March 25, 2014 Pages

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

Objective - To factor trinomials in the form,

Notes Over 10.8 Methods of Factoring Binomial Trinomial

Entry Task What is the polynomial function in standard form with the zeros of 0,2,-3 and -1?

Polynomials and Polynomial Functions

Polynomials & Factoring

Module 3.3 Factoring.

Section 6.4: Factoring Polynomials

Polynomial Equations and Factoring

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

4.5 & 4.6 Factoring Polynomials & Solving by Factoring

Factoring By Grouping and Cubes.

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

Factoring Polynomials

WARM UP Factor the polynomial completely. b b2

Objective - To factor trinomials in the form,

Polynomials and Polynomial Functions

Warm-Up 5 minutes List all the factors of each number. 1) 10 2) 48

Factor and Solve Polynomial Equations Lesson 2.4

5.4 Factor and Solve Polynomial Equations

Factor & Solve Polynomial Equations

Factor Special Products

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)

Objective - To factor trinomials in the form,

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

Unit 2 Algebra Investigations

Presentation transcript:

6.5 Factoring Cubic Polynomials 2/13/2013

Perfect Cubes 1000 = 103 729 = 93 512 = 83 343 = 73 216 = 63 125 = 53 64 = 43 27 = 33 8 = 23 1 = 13

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

Example 1 a. Factor . x 3 + 64 b. Factor . 8p 3 – q 3 SOLUTION x 3 + Factor the Sum or Difference of Two Cubes a. Factor . x 3 + 64 b. Factor . 8p 3 – q 3 SOLUTION Write as sum of two cubes. x 3 + 64 = 43 a. ( ) 4 x + x 2 4x – 42 = Use special product pattern. ( ) 4 x + x 2 4x – 16 = Simplify.

[ ] Example 1 b. 8p 3 – q 3 ( ) 2p 3 = = – ( ) q 2p + q2 2pq 2p 2 = – Factor the Sum or Difference of Two Cubes b. 8p 3 – q 3 ( ) 2p 3 = Write as difference of two cubes. = – ( ) q 2p + q2 2pq [ ] 2p 2 Use special product pattern. = – ( ) q 2p + q2 2pq 4p2 Simplify. 5

Checkpoint Factor the polynomial. ANSWER ( ) 1 x + x 2 – 1. x 3 + 1 2. 125x 3 + 8 ( ) 2 5x + 25x 2 10x – 4 3. x 3 216 – ( ) 6 x + x 2 6x 36 –

Finding Greatest Common Factor (GCF) Find the GCF of the terms in the polynomial: 𝑥 3 +𝑥 2 3 𝑥 3 −18 2 𝑥 3 +8 𝑥 2 −12𝑥 𝑥 2 3 2x

Example 2 a. Factor x 3 5x 2 6x. + – b. Factor 16x 4 2x. – SOLUTION Factor Polynomials a. Factor x 3 5x 2 6x. + – b. Factor 16x 4 2x. – SOLUTION Factor common monomial. x 3 5x 2 6x + – = ( ) x x 2 5x 6 a. Factor trinomial. = x ( ) 3 – 2 Factor common monomial. = ( ) 2x 8x 3 1 – b. 16x 4 Use special product pattern. = ( ) 2x 1 – 4x 2 + 8

Checkpoint Factor the polynomial. ANSWER x ( ) 1 – 3 + 4. x 3 + 2x 2 – Factor Polynomials Factor the polynomial. ANSWER x ( ) 1 – 3 + 4. x 3 + 2x 2 – 3x 5. 2x 3 10x 2 8x – + 2x ( ) 4 x – 1 6. 3x 4 24x + 3x ( ) 2 x + x 2 2x 4 – 7. 54x 4 16x – 2x ( ) 2 3x 9x 2 6x 4 + –

Homework: WS 6.5 #1-18 skip #8

Example 4 Factor the polynomial. a. x 2 ( x – 1 ) – 9 ( x – 1 ) b. x 3 Factor by Grouping Factor the polynomial. a. x 2 ( x – 1 ) – 9 ( x – 1 ) b. x 3 – 2x 2 – 16x + 32 SOLUTION Use distributive property. a. x 2 ( ) 1 x – 9 = Difference of two squares = ( ) 3 x – + 1 11

Example 4 b. = ( ) x 3 – 32 2x 2 16x + = ) x 2 – ( 2 + 16 x = ) – 16 ( Factor by Grouping b. = ( ) x 3 – 32 2x 2 16x + Group terms. Factor each group. = ) x 2 – ( 2 + 16 x Use distributive property. = ) – 16 ( 2 x x 2 Difference of two squares = ( ) 4 x – + 2 12

Factor the polynomial by grouping. ANSWERS Checkpoint Factor by Grouping Factor the polynomial by grouping. ANSWERS ( ) 2 x – + 6 8. x 2 ( ) 6 x + 4 – 9. x 3 4x 2 25x – 100 + ( ) 5 x – + 4 10. x 3 3x 2 4x 12 + ( ) 3 x + 4 x 2

Example 5 Solve 2x 3 14x 2 – = 24x. SOLUTION 2x 3 14x 2 – = 24x + ( ) Solve a Cubic Equation by Factoring Solve 2x 3 14x 2 – = 24x. SOLUTION 2x 3 14x 2 – = 24x + Rewrite in standard form. ( ) x 2 7x – = 12 + Factor common monomial. 2x ( ) 4 x – 3 2x = Factor trinomial. Use zero product property. 4 x – 3 2x = or x = 0, 4, 3 Solve for x. 14

Example 6 Solve x 3 6x 2 – = 2x. + 12 SOLUTION x 3 6x 2 – = + 12 2x ( Solve a Cubic Equation by Factoring Solve x 3 6x 2 – = 2x. + 12 SOLUTION Rewrite in standard form. x 3 6x 2 – = + 12 2x ( ) x 3 6x 2 – = 12 + Group terms. 2x ( ) 6 x – x 2 = Factor each group. + 2 ( ) 6 x – = Use distributive property. 2 x 2 6 x – = Use zero product property. 2 x 2 or +2 +2 +6 +6 x2 = 2 x = 6 15

Solve the equation by factoring. ANSWER Checkpoint Solve a Cubic Equation by Factoring Solve the equation by factoring. ANSWER 4, 0, 1 – 13. x 3 + 3x 2 = 4x 14. 3x 3 30x – = 9x 13, 0 + – 15. x 3 2x 2 3x + – 6 = 2, – 3 + 16. x 3 7x 2 5x = – 35 5, 7 + –