Algebra I Notes Section 9.6 (A) Factoring ax 2 + bx + c With Leading Coefficient ≠ 1.

Slides:

Advertisements

Similar presentations

Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz.

Advertisements

7-3 Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz

Factoring Trinomials of the form

1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives Factoring Trinomials Factor trinomials when the coefficient of the quadratic term.

7-3 Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz

Factoring Trinomials of the form x 2 + bx + c Chapter 5.3.

CRASH COURSE IN QUADRATICS In preparation for the Algebra CST -b + b 2 – 4ac 2ac √ (x+4)(x-3)=0 (x+1)(x+2) X 2 – 5x +4 F O I L Complete The Square.

Polynomials and Polynomial Functions

7-3 Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz

4.5 Multiplying Polynomials

Essential Question: How is FOIL related to factoring?

Binomials. What is a binomial?  A binomial expression is an expression with 2 terms.  EXAMPLES: x+2, 2p-3, p+q.

§ 4.5 Multiplication of Polynomials. Angel, Elementary Algebra, 7ed 2 Multiplying Polynomials To multiply a monomial by a monomial, multiply their coefficients.

8-4 Factoring ax2 + bx + c Warm Up Lesson Presentation Lesson Quiz

Exponents and Polynomials

EXPONENTS AND POLYNOMIALS College Algebra. Integral Exponents and Scientific Notation Positive and negative exponents Product rule for exponents Zero.

1. The height of an object launched t seconds is modeled by h(t) = -16t t Find the vertex and interpret what it means. What is the height of.

Objective 1.Factor quadratic trinomials of the form x2 + bx + c.

Algebra I Notes Section 9.5 (A) Factoring x 2 + bx + c With Leading Coefficient = 1 To factor a quadratic expression means to write it as a product of.

1 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-1 Polynomials and Polynomial Functions Chapter 5.

Factoring Trinomials. Recall by using the FOIL method that F O I L (x + 2)(x + 4) = x 2 + 4x + 2x + 8 = x 2 + 6x + 8 To factor x 2 + bx + c into (x +

Algebra Choosing a Factoring Method. Learning Targets Language Goal: Students will be able describe an appropriate method for factoring a polynomial.

Section 4.4 – Factoring Quadratic Expressions Factors of a given number are numbers that have a product equal to the given numbers. Factors of a given.

Holt Algebra Factoring x 2 + bx + c Factor quadratic trinomials of the form x 2 + bx + c. Objective.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1 1 Chapter 6 Factoring.

Preview Warm Up California Standards Lesson Presentation.

Chapter 7 Section 3 and 4 Factoring. Objectives  Factor quadratic trinomials of the form x 2 + bx + c. Factor quadratic trinomials of the form ax 2 +

Welcome to MM218! Kirsten Meymaris, Mar 15 th Unit 3 : Factoring Part 2 Plan for the hour Review of Factoring from Unit 2 MML questions from Unit 2 Test.

Factoring Trinomials Module VII, Lesson 5 Online Algebra

Factoring Easy and Hard Trinomials MATH 017 Intermediate Algebra S. Rook.

Factoring and Solving Polynomial Equations (Day 1)

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

Chapter 5.2 Solving Quadratic Equations by Factoring.

REVIEW OF FACTORING Chapters 5.1 – 5.6. Factors Factors are numbers or variables that are multiplied in a multiplication problem. Factor an expression.

Section 10.6 Factoring Objectives: Factor a quadratic expression of the form Solve quadratic equations by factoring.

Chapter 5 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-1 Factoring.

Chapter 5 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-1 Factoring.

Holt McDougal Algebra 1 Factoring x 2 + bx + c Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz Holt McDougal.

Factoring – Day 4 Factoring Trinomials Objective: To factor trinomials whose quadratic coefficient is 1.

Multiplying Polynomials with FOIL Objective: Students will multiply two binomials using the FOIL method. S. Calahan March 2008.

Factoring Example 1: What is the Greatest Common Factor (GCF) of the two terms below? Example 2: Example 3:

Holt McDougal Algebra 1 Factoring x 2 + bx + c Factor quadratic trinomials of the form x 2 + bx + c. Objective.

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 +

Factoring Trinomials Chapter 10.4 Part 2. Review: Factoring Quadratic Trinomials Find the factors of the last term. Which of those factors combine to.

Holt McDougal Algebra Factoring x 2 + bx + c Factor quadratic trinomials of the form x 2 + bx + c. Objective multiply two binomials using the Distributive.

MTH Algebra Factoring Trinomials of the form ax 2 + bx + c where a = 1 Chapter 5 Section 3.

Factoring Quadratic Trinomials a = 1 Chapter 10.5.

Factoring Quadratic Expressions Lesson 4-4 Part 1

Multiply two binomials using FOIL method

7-3 Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz

Factoring Review Algebra 2.

Section R.4 Factoring.

Chapter 6 Section 2.

Warm Up Find each product. 1. (x – 2)(2x + 7) 2. (3y + 4)(2y + 9)

8-3 Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz

7-3 Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz

8-3 Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz

Objective Factor quadratic trinomials of the form ax2 + bx + c.

Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz

7-3 Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz

Algebra 1 Section 10.3.

Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz

1) Which pair of factors of 8 has a sum of 9?

Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz

7-3 Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz

7-3 Factoring x2 + bx + c Warm Up Lesson Presentation Lesson Quiz

2.3 Factor and Solve Polynomial Expressions

There is a pattern for factoring trinomials of this form, when c

Presentation transcript:

Algebra I Notes Section 9.6 (A) Factoring ax 2 + bx + c With Leading Coefficient ≠ 1

In section 9.5, we learned how to factor quadratic polynomials whose leading coefficient = 1. We will now learn how to factor quadratic polynomials whose leading coefficient is ≠ 1. To factor quadratic polynomials when a ≠ 1, we need to find the factors of a and the factors of c such that when we add the Outer and Inner products of the FOIL method, their sum = b. Example – Factor the polynomialsWe must find the factors of a and c 1.3x 2 – 17x the factors of 3 (or a) are: ___________ - the factors of 10 (or c) are : ___________ We know that the binomials (without their signs) must be one of these four forms because of the factors: , 3 1, 10 & 2, 5 (x 1)(3x 10)(x 10)(3x 1)(x 2)(3x 5)(x 5)(3x 2)

If we look at the FOIL method for each of the possible factors, we get: F O I L We know that in the original polynomial 3x 2 – 17x + 10, 10 (or c) must be positive, so the L multiplication of the FOIL method must yield a + 10 (for c). At the same time, the sum of the O and I multiplication must yield a - 17 (for b). With the correct choice of signs, which sum (of O and I) can possibly add up to -17? _______________ So we conclude: 3x 2 – 17x + 10 = ____________________ 3x 2 10x3x10 3x 2 x30x10 3x 2 5x6x10 3x 2 2x15x10 -2x & -15x (x – 5)(3x – 2)

2. 3x 2 – 4x – 7- the factors of 3 (or a) are: _____________ - the factors of 7 (or c) are : _____________ We know that the binomials (without their signs) must be one of these two forms because of the factors: If we look at the FOIL method for each of the possible factors, we get: F O I L We know that in the original polynomial 3x 2 – 4x – 7, -7 (or c) must be negative, so the L multiplication of the FOIL method must yield a -7 (for c). At the same time, the sum of the O and I multiplication must yield a - 4 (for b). With the correct choice of signs, which sum (of O and I) can possibly add up to -4? ________________ So we conclude: 3x 2 – 4x – 7 = ________________ 1, 3 1, 7 (x 1)(3x 7)(x 7)(3x 1) 7x3x x21x -7x & 3x (x + 1)(3x – 7)

3. 6x 2 – 2x – 8- begin by factoring out the common factor ___________ - the factors of 3 (or a) are: _______________ - the factors of 4 (or c) are : _______________ We know that the binomials (without their signs) must be one of these three forms because of the factors: If we look at the FOIL method for each of the possible factors, we get: FOI L We know that in the FACTORED polynomial 3x 2 – x – 4, -4 (or c) must be negative, so the L multiplication of the FOIL method must yield a -4 (for c). At the same time, the sum of the O and I multiplication must yield a - 1 (for b). 2 1, 3 2(3x 2 – x – 4) 1, 4 & 2, 2 (x 1)(3x 4)(x 4)(3x 1)(x 2)(3x 2) 4x3x x12x 2x6x With the correct choice of signs, which sum (of O and I) can possibly add up to -1? ________________ So we conclude: 2(3x 2 – x – 4) = _______________________ -4x & 3x 2(x + 1)(3x – 4)

More Examples - Factor the polynomials. 1.3t t b 2 – 11b – 2 Factors of a : 1, 3 Factors of c : 1, 5 (t 1)(3t 5)(t 5)(3t 1) 5t, 3tt, 15t Must add to 16t : (t + 5)(3t + 1) Factors of a : 1, 6 2, 3 Factors of c : 1, 2 (2b 1)(3b 2)(2b 2)(3b 1) 4b, 3b2b, 6b (b 1)(6b 2)(b 2)(6b 1) 2b, 6bb, 12b Must add to -11b : (b – 2)(6b + 1)

3. 5w 2 – 9w – 24. 4x 2 – 6x – 4 Factors of a : 1, 5 Factors of c : 1, 2 (w 1)(5w 2) (w 2)(5w 1) 2w, 5w w, 10w Must add up to -9w : (w - 2)(5w + 1) Factors of a : 1, 2 Factors of c : 1, 2 Factor out GCF : 2 2(2x 2 – 3x – 2) (x 1)(2x 2) (x 2)(2x 1) 2x, 2x x, 4x Must add up to -3x : 2(x – 2)(2x + 1)

5. 12y 2 – 22y – x x – 18 Factor out GCF : 2 2(6y 2 – 11y – 10) Factors of a : 1, 6 2, 3 Factors of c : 1, 10 2, 5 (2y 2)(3y 5) (2y 5)(3y 2) 10y, 6y 4y, 15y Must add up to -11y : 2(2y – 5)(3y + 2) Factor out GCF : 3 3(3x 2 + 7x – 6) Factors of a : 1, 3 Factors of c : 1, 6 2, 3 (x 2)(3x 3) (x 3)(3x 2) 3x, 6x 2x, 9x Must add up to 7x : 3(x + 3)(3x – 2)