In my experience teaching this class -- CHAPTER 4 IS THE KEY TO SUCCESS.  Everything we have done up to this point is about preparing us to work with factoring polynomials (Chapter 4).  Students that have developed some comfort, confidence and accuracy with the previous 4 chapters (R-3) will be able to complete this chapter.

In many way; Chapter 5, Rational Expressions revisits Chapter 4 and extends student learning.  It is rare to see a student struggle with Chapters 5, 6, 7, 8, and 9 if they have mastered Chapter 4.  Please apply yourself fully to this unit -- in my humble opinion, it is the key to Success in Beginning Algebra 41.  It will also be the key to success starting Algebra 141 with a solid foundation of skills to build on.

4.1 The Greatest Common Factor

In many ways, factoring is the REVERSE of multiplications.  2*3=6.  Two factors of 5 are 2 and 3.  The process of writing a polynomial as a product is called factoring.

5x2+25x is a binomial (a polynomial with 2 terms.  We can write this binomial as a product:  5x(x+5).  This is an example of factoring. 

Suppose we had started with the product 5x(x+5) -- when we perform that multiplication (distributed property), we get 5x2+25x.   Do you see why factoring is the reverse of multiplication?

Definitions:

Factoring:  process of writing an expression as a product.

Greatest common factor (GCF) of a list of common variables raised to powers is the variable raised to the smallest exponent in the list (please refer to examples, page 271 in text).

Greatest common factor (GCF) of a list of terms is the product of all common factors (please refer to examples, pages 272-273 in text).

Factor by Grouping:

  1. Group terms into 2 groups of 2 terms each.

  2. Factor out the GCF from each group.

  3. If there is a common binomial factor, factor it out.

  4. In not, rearrange the terms and try steps 1-3 again.

Please see examples pages 273-174 in text.

4.2 Factoring Trinomials in form x2+bx+c

In this section, we will look at trinomials that have a "lead" coefficient of "1."  Note that we do not write coefficients of 1, they are assumed.

Factoring is writing as a project, remember that multiplication is commutative -- we can factor the trinomial x2+7x+10 as either (x+2)(x+5) or (x+5)(x+2).  To see why, multiply each of those 2 factors (FOIL) and you will get the original expression when you write the answer in descending order.

The key to factoring trinomials in the form x2+bx+c is to remember that we will end up with a product of 2 binomials (x+?1) and x+?2) where ?1 and ?2 are numbers whose project is c.  Please see examples, pages 280-281.

Note how the signs form a pattern.  A positive constant in a trinomial (c) means to look for 2 numbers (?1 and ?2) that have the same sign (positive and positive or negative and negative).  The sign of the coefficient of the middle term tells us whether the signs are both positive or both negative.

A negative constant in a trinomial means to look for 2 numbers with opposite signs.  Remember, that is the only way that the result of our product can be negative.  Please see examples, page 282.

Factoring Out the Greatest Common Factor.

ALWAYS REMEMBER that the first step in factoring any polynomial is to factor out the greatest common factor (if other than 1 or -1).  DO NOT FORGET that you have pulled out a common factor when this is possible.  YOUR FINAL ANSWER MUST INCLUDE THIS GCF.  Please see examples, page 282.

Factoring Trinomials in the Form ax2+bx+c

Hopefully, starting with trinomials in the form x2+bx+c allowed us to start looking at factoring patterns by considering the simplest case -- "a" or the lead coefficient equaling 1.

Page 287 illustrate the process of rewriting ax2+bx+c as a product using "guess and by golly."  In the examples illustrated, the correct answer is always presented as the "last guess."  Some may find this presentation tedious or confusing, but it is correct -- the author has just decided to show us all of the WRONG factors first and then presents us with the correct factor.   

WITH PRACTICE, STUDENTS WILL START RECOGNIZING THE CORRECT FACTORS AND NOT HAVE TO LOOK AT EACH WRONG FACTOR!  Please study the examples, page 287-288, but don't let the author's presentation intimidate you.

Try factoring trinomials in the form ax2+bx+c by finding combinations of factors of ax2 and c until the middle term of bx is obtained when checking.  Remember, always pull out any GCF first and be sure to include GCF with your final answer.  If a trinomial has no GCF, then the terms of its binomial factor will not contain a common factor (other than 1) either.

Please review examples, pages 288-190.  Do not let the presentation bother you -- the author will show us all the ways that DON'T work before she shows us the one that DOES work.  

With practice, you will not look at every situation that does not work -- you will start seeing the patterns that do work.  There is another way to factor trinomials -- that is presented the next section, but it is based on the factoring by grouping that we discussed at the beginning of this unit.

Factoring Trinomials in the Form ax2+bx+c by Grouping

Another method that can be used is to apply the factoring by grouping or grouping method we looked at earlier.   Some treat this as a great "secret" that makes factoring simple -- I am sure that it will be very beneficial to some students.  IT DOES NOT MATTER WHAT METHOD WE USE TO FACTOR TRINOMIALS AS LONG AS WE GET AN ANSWER THAT CORRECTLY  CHECKS.

In my experience teaching this class, students that practice the problems and procedures in section 4.3 will start to see patterns in numbers and will be able to factor trinomials in the form ax2+bx+c by realizing those factors that will work. 

Many of those students will find section 4.4 a bother -- they do not need another method to factor trinomials.  Some students, however, will find that the grouping method makes more sense, especially if they they did not find the presentation in chapter 4.3 helpful.  

I am going to ask students to look at both and apply the method that works best for them.  I will not require anyone to use one method or the other.  Please look at each of these methods and decide what one will be most comfortable for you. 

Factoring Trinomials of the form ax2+bx+c by grouping:

  1. Factor out a greatest common factor, if there is one other than 1 (or -1)

  2. Find 2 numbers whose product is a*c and whose sum is b.

  3. Write the middle term, bx using the factors from step 2.

  4. Factor by grouping (method illustrated in section 4.1).

Please review the examples, pages 295-296.  Remember, you may choose to factor trinomials in the form ax2+bx+c by the method that you are most comfortable with.

Factoring Perfect Square Trinomials and the Difference of Two Squares

Remember the "special products" in chapter 3 -- our author stressed them in that part of the book to prepare us to factor trinomials.  Recall the special product formulas for squaring binomials:  (a+b)2 = a2+2ab+b2 and (a-b)2=a2-2ab+b2.

A trinomial that is a square of a binomial is called a perfect square trinomial, because x2+6x+9 factors into (x+3)2, it is a perfect square trinomial.

Likewise, 4x2-12x+4 is a perfect square trinomial, because it factors into (2x-3)2 .

Remember you can factor trinomials by EITHER of the methods covered in 4.2 or 4.4.  It is important to recognize these special cases, which are really just extensions of the special products we looked at in Chapter 3.  Seeing these patterns will save you time and make the Chapter 5, rational expressions, much easier.  Review the examples, pages 301-303.

Factoring the Difference of 2 Squares

Recall from Chapter 3 that we looked at anothwer special product, the product of the sum and difference of 2 terms a and b:  (a+b)(a-b)=a2-b2.

Just like we applied the special products rules to factor trinomials, we can reverse the equation above to create another factoring pattern which will be used to factor the difference of 2 squares

Factoring the difference of squares:  a2-b2=(a+b)(a-b).  Please review the examples, pages 303-304

Don't forget that we must always check to see if a trinomial contains a greatest common factor (GCF) BEFORE trying any other factorization method.  

ALWAYS BE SURE TO FACTOR COMPLETELY!  Check to see if any factors can be factored further.  Our textbook author is going to use many examples where a trinomial factors with one term being the difference of 2 squares (exampler 15, page 304).  

Watch for that situation!  She is trying to help us prepare for Chapter 5, Rational Expressions.  If you practice the problems and develop some comfort with these examples, you will do fine in the chapters that follow.

4.6 Solving Quadratic Equations by Factoring

In many ways, this section is a review of the previous section (4.5).  Please apply yourself fully to this unit.  It will review and reinforce the main concepts in the chapter and prepare you for Chapter 5.

If we take a trinomial in the form of ax2+bx+c and set it equal to zero, it is a quadratic equation.  Remember, x2+bx+c is just a special case where a=1.

Quadratic Equation:  an equation that can be written in the form ax2+bx+c=0 where a, b, and c are real numbers and a does not equal 0.

Note that our definition DOES allow b and c to equal zero, hence we can have quadratic equations with only our lead term.  Please review the examples on page 313.

If a quadratic equation is factorable (remember, not all trinomials are factorable), then we can solve it by applying the zero factor property.

Zero Factor Property:  If a and b are real numbers and if ab=0, then a=0 or b=0.

This is not new -- we have looked at this before.  This understanding tells us that if the product of 2 numbers is 0, then at least one of the numbers must be 0.  Please look at these examples:

bullet

If (a*b)=0, then a or b MUST BE 0.

bullet

If x(x+5)=0, then x=0, or x+5=0.

bullet

If (x+7)(2x-3)=0, then x+7=0 ir 2x--3=0.

Please review the examples, pages 314-315 to see the different ways we need to be able to apply the zero factor property.

To Solve Quadratic Equations by Factoring:

  1. Write the equation in standard from so that one side of the equation is 0

  2. Factor the quadratic equation completely.

  3. Set each factor containing a variable equal to 0

  4. Solve the resulting equations

  5. Check each solution in the original equation

Please review the examples, pages 316-317.

4.7 Quadratic Equations and Problem Solving

This unit shows us some examples of how problems can be solved with the quadratic equation -- it serves as a great chapter review and illustrates some of the reasons that understanding algebra is an important skill for college students to master.  

Section 4.7 is all about critical thinking and applying the steps of problem solving:

  1. Understand problems

  2. Translate

  3. Solve

  4. Interpret

The only "new" concepts in section 4.7 are:

bullet

When solving quadratic equations, we are not done just because we have solved for the value x -- our final answer must actually answer the question we started out with.  

bullet

While we may algebraically determine that a number of solutions are possible, only some of those possible solutions may be practical.  Always make sure that proposed solutions make sense when applied to the problem at hand -- remember, we must actually answer the question we started with and any proposed solution must make sense in the context of what is being asked.

Please review the 4 examples, pages 324-327.  Not only do the review important concepts, but they illustrate problem-solving.

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Class Notes

Review Notes

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Chapter 2 Notes

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Chapter 4 Notes

Chapter 5 Notes

Chapter 6 Notes

Chapter 7 Notes

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