Chapter 2 Para”boo”la created entirely out of parabolas (graphs of quadratic polynomials), whose translations can be described when in factored form.

In chapter 1, we were introduced to polynomials.

From exploring, we discovered polynomials could be added or subtracted together to form other polynomials, by simply adding or subtracting like terms.

We also discovered polynomials could be multiplied by each other to form other polynomials, but unfortunately, just multiplying like terms wasn’t enough.

We found to multiply polynomials together, we had to use the distributive property, and multiply all of the terms in one polynomial by every term in the other polynomial. We also discovered that the degree of the product polynomial was the sum of the degrees of the two polynomials being multiplied. For example, a linear (degree 1) polynomial times another linear (degree 1) polynomial, resulted in a quadratic (degree 2) polynomial.  Graphically speaking, when we multiplied two polynomials whose graphs were both nonhorizontal lines, the result was a polynomial whose graph was a parabola.

This chapter is about reversing that process and being able to factor polynomials.  The same way that all positive integers (greater than 1) can be factored uniquely into prime numbers, all polynomials (with degree greater than 1) can be factored uniquely into linear (degree 1) polynomials (times the GCF’s of the terms of the original polynomials).

 For example, the prime factorization of 12, is ( 2 )( 2 )( 3 ). Similarly, the polynomial  2x3-13x2+8x+48,can be factored into (x – 4)(x – 4)(2x + 3).

In the above example, the linear (degree 1) polynomial factors are “prime like” because they cannot be factored further.  As you may have noticed, that doesn’t mean the coefficients and constants have to be prime numbers.  In fact, they can be any kind of number, even radicals or complex numbers.

As a result of the Fundamental Theorem of Algebra, all polynomials (with degree greater than 1) can be uniquely factored into linear (degree 1) polynomials.  That’s too much to handle at this point, so for now, we will focus on factoring quadratic (degree 2) polynomials whose graphs are parabolas, into two linear (degree 1) polynomials whose graphs are lines. For a visual overview of what you have just read, run the applet  “multlinpolys.html”  , and move the chapter slider.  After seeing the connection between the two questions from each chapter, go on to the next page.

Given that every quadratic (degree 2) polynomial can be uniquely factored into two linear (degree 1) polynomials, our challenge is to figure out how.  The idea of factoring any degree 2 polynomial, is still too overwhelming at this point however.  If you need to be convinced before continuing, try the following example by guessing and checking (write down two degree 1 polynomial factors, and then multiply them together using the distributive property to check your guess).

 What are the degree 1 polynomial factors of  9x2 – 12x – 41?

A common strategy in math, is to first study special cases (specific types of examples).  This chapter will focus on examples, where the coefficients and constants of the degree 1 polynomial factors are integers.  Examples like the one above, will be skipped for now, because the factors require the use of radicals.

Actually, our first challenge will be to discover how to factor monic degree 2 polynomials.  Monic polynomials have lead coefficients of 1.  In other words, our first challenge will be to discover how to factor degree 2 polynomials that look like the following examples.

 1x2 – 8x + 12                    1x2 – 49                    x2 + 8x + 7                    x2 – 4x – 60

Notice, the x squared coefficients in each example above have a value of 1 (whether the 1 is written or not).

 In case you have to know,  9x2 – 12x – 41  factors into  ( 3x – [ 2 – 3√5 ] )( 3x – [ 2 + 3√5 ] )

After chapter 4, we will be able to figure out how the above factors were found.  For now though, the only polynomials that we will consider factorable, are the ones that have factors with both integer coefficients and integer constants.

Our first challenge of chapter 2 will be to discover the pattern behind factoring monic degree 2 polynomials.  To find a pattern though, we will need examples.  Part of the challenge however, will be to create our own examples, by using an area model for the problems. Given lengths of 3 units and 4 units, the only possible area for the rectangle is 12 square units.  Going the other way isn’t as straightforward however.  Given a rectangle with an area of 12 square units, the dimensions of the rectangle could have also been 2 units by 6 units, or 1 unit by 12 units.  Composite numbers have more than one pair of factors.

An area model can be used for multiplying degree 1 polynomials as well.  In the example below, the degree 1 polynomials  ( x + 3 ) and ( x + 4 ) are represented by the length and width of the rectangle, and the degree 2 polynomial  x2 + 7x + 12  is represented by the area of the rectangle. Just like the area model for 3 • 4 = 12, the only possible area for a ( x + 3 ) unit by ( x + 4 ) unit rectangle, is  x2 + 7x + 12 square units.  Unlike the area model for  3 • 4 = 12  however, a rectangle with an area of  x2 + 7x + 12  square units, can only have sides with lengths of ( x + 3 ) units and ( x + 4 ) units.  Monic degree 2 polynomials have only one possible pair of degree 1 factors.

Below are snapshots of an applet, which will allow you to explore area models for monic degree 2 monic polynomials with positive coefficients and constants.  Run the applet  “areamodel.html”, create an area model for at least 3 quadratics, and answer the resulting question. Use a copy of the pdf  appletnotes  to record your answer, and defend it with at least 3 examples. After recording and defending your answer with at least 3 examples, go on to the next page.

An area model can work for monic degree 2 polynomials, with subtraction involved as well.  We just start with a blue square with an area of x2 square units, and subtract the areas of the rectangles that were taken away. Another way of looking at it is, a square with an area of x2 square units, had 5 rectangles with areas of x square units taken away from it.  But that would mean, the 4 squares in the upper right hand corner with areas of 1 square unit each, would have been taken away twice.  So, we put the second set of 4 squares with areas of 1 square unit each back.

An area model can work for monic degree 2 polynomials, even if both addition and subtraction is involved.

A square with an area of x2 square units had 5 horizontal rectangles with areas of x square units each taken away. At the same time though, 6 vertical rectangles with areas of x square units each were added. The vertical rectangles were too tall however, so 30 squares with areas of 1 square unit each had to be taken back.  Run the applet  “areamodel2.html”  , create area models for at least 3 quadratics, and answer the resulting question. Use a copy of the pdf  appletnotes, to record your answer, and at least 3 examples involving negatives. After recording and defending your answer, with at least 3 examples involving negatives, go on to the next page.

Due to the fact that actual distances and areas are always positive, area models only makes sense if the input values for x are positive, and the output values of the polynomials are positive as well.  As mentioned before, using geometry is a great way to make specific ideas more concrete.

However, even if negative input values for x are used and/or the output values for the polynomials end up negative, the pattern we found using the last 2 applets will still work algebraically, even though the area models won’t make sense. For example, using an input value of -3 doesn’t change the fact that the pattern we found works to factor x2 + 4x – 12 into ( x + 6 )( x – 2 ), even though it’s impossible to have a rectangle with a side -5 units long and an area of -15 square units. That’s the power of algebra, it can generalize and expand ideas, and help lead to a deeper understanding of those ideas, whether they are abstract or not.

Before continuing, it should be pointed out, that geometry can be quite abstract and have nothing to do with the physical world.  Even things as simple as points, lines, and rectangles are abstract.  These are purely mathematical objects and don’t actually exist in real life.  A point is infitesimally small, but the things that make up atoms are not, so it would be impossible to create a point.  Lines are infitesimally long, but our universe isn’t, so it would be impossible for a line to exist.  Lines are also infitesimally narrow, but again, the things that make up atoms are not, so it would be impossible to create a line even if the universe was infinite in size.  Rectangles have perfectly straight edges and perfectly flat surfaces, but looking at any real object under a microscope would reveal a surface that is quite uneven, so it would be impossible to create anything perfectly straight or flat.  Even if it were possible to create these objects, it would be impossible for us to see infitesimally small points, infitesimally narrow lines, or the infitesimally narrow line segment edges of rectangles.  The geometric representations of these mathematical objects which we draw, are good enough for practical purposes though.  Using geometry to make ideas more concrete, actually refers to using the geometric representations of geometric objects.

To test the pattern we found for factoring monic quadratics into two linear factors,  we will now switch to a different geometric model for polynomials – their graphs.  The graphs of quadratics are parabolas and the graphs of their linear polynomial factors are lines, both of which have no problem with negative input or output values.  Again, parabolas and lines are purely abstract mathematical objects.  The graphs we will be using are just practical geometric representations of these objects. In the example to the right, the black parabola is the graph of the quadratic  x2 – 9x – 22  , which we would like to factor into two linear polynomial factors.

The gold lines are the graphs of the linear polynomial factors ( x + 15 ) and ( x – 15 ) , and the gold parabola is the graph of their product. Answer the following two questions.

1)  Does  x2 – 9x – 22  factor into  (x + 15)(x – 15) ?

2)  What would be true of the two parabolas if x2 – 9x – 22  did factor into  (x + 15)(x – 15)? After answering the questions above, go on to the next page.

Below is a snapshot of an applet, which will allow you to further explore factoring monic quadratic (degree 2) polynomials. The challenge is to factor the quadratic shown in black, using the pattern we found from exploring the area models for factoring monic quadratics.

The product of the linear factors, is represented by the parabola shown in gold.

Trinomials are polynomials which have 3 terms, and binomials are polynomials which have 2 terms. Run the applet  “factorquadsgraph1.html”  , until you can factor at least 3 quadratics, without using the graphs. Use a copy of the pdf  appletnotes to record the quadratics, the factors, and any discoveries about the graphs. After factoring at least 3 quadratics, with the 3 boxes unchecked, and recording your work, go on to the next page.

Below is a snapshot of a program, which will randomly generate 12 monic quadratics for you to factor, without geometric representations.  Run  program 201  until you can factor at least 10 of the 12 monic quadratic polynomials. Use a copy of the pdf  programnotes  to record the quadratics, any work involved, and the linear factors. After factoring at least 10 of the quadratics, and neatly recording all of your work, go on to the next page.

Our next challenge, is to now try to find a pattern for factoring degree 2 polynomials when the x squared coefficient is greater than 1.  Below are snapshots of an area model program which will randomly generate quadratics for you to factor.  The program will stop after you factor 5 problems correctly.  It will then show you a list of the 5 problems along with your answers.  By looking at this list, your challenge is to find a pattern between the quadratics and their two linear factors.  Run  program 202  , and use the area models until you can correctly factor 5 quadratics, and then discover a pattern between the quadratics and their two linear factors. Use a copy of the pdf  programnotes, to record the 5 quadratics, their factors, and the pattern you discover. After neatly recording the 5 quadratics, their factors, and the pattern which connects them , go on to the next page.

Again, an area model only makes sense for positive distances and areas.  We’ll now switch to the graphs of the polynomials as our geometric representation model, to see if the pattern we just found, still works with negative values.

Below are snapshots of the next applet to be used for exploration.  The challenge will be to factor the given quadratic into two linear factors, by moving the sliders.  As we now know, the given quadratic and its factored expression are equivalent functions.  In other words, if the same input value is used, both expressions will produce the same output value.  This means the graphs of both expressions will be identical after factoring correctly.  Run the applet  “factorquadsgraph2.html”  , until you can factor at least 3 quadratics, without using the purple graphs. Use a copy of the pdf  appletnotes  to record the 3 quadratics, their factors, and any discoveries. After factoring and recording at least 3 quadratics, with both boxes unchecked, go on to the next page.

At this point, you have found patterns for factoring both monic and nonmonic quadratics.

 monic quadratic:     x2 + 5x + 6  =  ( x + 2 )( x + 3 ) nonmonic quadratic:   2x2 + 5x + 3  =  ( 2x + 3 )( x + 1 ) Answer the 3 questions below. Use a copy of the pdf  booknotes  to record and defend your answers.

1. Are the patterns you found for both types of quadratics, different or actually the same?
2. How would you answer question 1, if the two examples above were rewritten with all of the coefficients included?
1. 1x2 + 5x + 6  =  ( 1x + 2 )( 1x + 3 )
2. 2x2 + 5x + 3  =  ( 2x + 3 )( 1x + 1 )
3. How are the patterns you found related to the distributive property? After neatly recording and defending your answers to the above 4 questions, go on to the next page.

Below are snapshots of a program, which will randomly generate 14 quadratics to factor, with lead coefficients greater than one.  Run  program 203  until you can factor at least 12 of the 14 quadratics, without using geometric representations. Use a copy of the pdf  programnotes  to record the quadratics, any work involved, and their linear factors. If you miss or skip, 3 or more problems while running program 203, open the pdf  FTA  for some hints. After factoring at least 12 out of 14 quadratics, and neatly recording all of your work, go on to the next page.

In general, a factor is a quantity by which we multiply, and a divisor is a quantity by which we divide. For example, in the multiplication problem  6 x 2 = 12 ,  2  is the factor, and in the division problem  12 ÷ 2 = 6 ,  2  is the divisor.

Most commonly though, the terms factors and divisors, are both used to refer to integers or polynomials that divide other integers or polynomials evenly.

For example, because  4 x 5 = 20 , the integers 4 and 5 are both factors and divisors of 20, and in ( x + 3 )( x – 7 )  =  x2 – 4x – 21 , the polynomials  ( x + 3 ) and ( x – 7 )  are both factors and divisors of  x2 – 4x – 21 .

That means, if there is no remainder involved, we can now divide a polynomial by another polynomial.

Due to the example above, we now know that  (x2 – 4x – 21) ÷ ( x + 3 )  =  x – 7 .

Since fractions are another way to represent division problems, we can now reduce fractions containing polynomials as well, when possible.  Reducing a fraction is just simplifying the fraction by reducing the number of common factors. Below are snapshots of a program, which will randomly generate 10 rational polynomial expressions (ratios of polynomial expressions in fraction form).  The challenge is to reduce them if possible.  Run  program 204  until you can reduce at least 8 of the 10 rational expressions. Use a copy of the pdf  programnotes to record the original and reduced expressions, and any work done. For more information and examples, open the pdf  rational expressions. After correctly reducing at least 8 of the rational expressions, neatly recording any work, and giving any restrictions to the domain (all of the allowable input values for x), go on to the next page.

So far, most of the focus has been on quadratics (degree 2 polynomials).  A polynomial of any degree can be factored if every term in that polynomial shares a common factor greater than one.

Below are snapshots of an applet which will randomly generate binomials (polynomials with 2 terms) for you to factor.  Run the applet  “factorgcf.html”  until you discover a pattern for factoring the binomials. Use a copy of the pdf  appletnotes  to record the pattern you discovered, and at least 3 examples. After discovering a pattern for factoring the binomials (when possible), and recording at least 3 examples, go on to the next page.

As we discovered earlier, factoring is just the distributive property in reverse.

Below are snapshots of a program which will randomly generate binomials to be factored, by taking out the GCF.  Run  program 205  until you can factor at least 10 of the 12 polynomials, by taking out the greatest common factor. Use a copy of the pdf  programnotes  to record the polynomials, their factors, and any work involved. After factoring out the GCF for at least 10 of the polynomials, and recording all of your work, go on to the next page.

In “program 205” , you were given linear (degree 1), quadratic (degree 2), and cubic (degree 3) binomials to factor. Below are snapshots of a program, which will randomly generate quadratic, cubic, and quartic (degree 4) trinomials (polynomials with 3 terms) to factor, by taking out the GCF.  Run  program 206  until you can factor at least 5 of the 7 polynomials, by factoring out the GCF. Use a copy of the pdf  programnotes  to record the polynomials, their factors, and any work involved. After factoring at least 5 polynomials, and recording all of your work, go on to the next page.

Now that we know how to factor polynomials by taking out the GCF, another way of factoring quadratic trinomials into linear binomials can be shared.  Notice in the snapshot below, steps 3 and 4 involve factoring by taking out the GCF.  To see if the above method is helpful, run the applet  “factorquads.html”  and/or open the pdf  factoring. Use a copy of the pdf  appletnotes  to record an example, the work involved, and any discoveries. After giving the above method a try, and recording an example, go on.

Now that we know how to factor polynomials by taking out the GCF (greatest common factor), a better idea of what mathematicians actually do, can be shared.  Mathematicians look for patterns and relationships between mathematical objects, make conjectures (statements believed to be true) about those patterns and relationships, and then try to create proofs (logical arguments that prove the conjectures to be true).

An example from the branch of mathematics called, number theory, will be used as an example.

Below are some examples of adding two odd integers together.

1 + 3 = 4

5 + 7 = 12

29 + 31 = 60

-19 + 9 = -10

-25 + 1 = -26

What do you notice about all of the answers? After answering the above question, go on.

A mathematician would notice that all of the answers to the examples were even integers.  After writing down a few more examples to support his or her hunch, a mathematician would make a conjecture (a statement believed to be true).

 Conjecture:  Given any two odd integers, the sum will always be an even integer.

Even if the mathematician wrote down hundreds of examples, and they all had even integer sums, that would not prove that the conjecture is always true. To prove it, the mathematician would have to show that it’s true for every possible sum of two odd integers.  Unfortunately, since there are an infinite number of odd integers, it would be impossible to check every possibility.  This is where polynomials and factoring step in to save the day.

 The linear polynomial 2n , where n is any integer, is a way to represent any and every even integer there is. No matter what integer n is, if you double it, you will have an even integer (even integers are multiples of 2).Odd integers are all of the integers in-between the even integers (they are all 1 away from an even integer). This means, all you have to do is add 1 to an even integer, and youʼll have an odd integer.So, the linear polynomial 2n + 1 , where n is any integer, is a way to represent any and every odd integer there is. The linear polynomial 2m + 1 , where m is any integer, is a way to represent any and every odd integer as well.

To show that any two odd integers will sum to an even integer, the mathematician would have to show that the polynomial representations of any two odd integers adds to a polynomial representation of an even integer.

In other words, the mathematician would have to show that the sum of two polynomial representations for any two odd integers, has the form of a polynomial representation for an even integer (a multiple of 2).  Try to prove the following conjecture.Conjecture:  Given any two odd integers, the product will always be an odd integer. Use a copy of the pdf  booknotes  to write your proof. After writing your proof, go on.

Conjecture:  The product of any two odd integers, will always be an odd integer. The dark square symbol (bottom right corner of image above) is used to show the end of a proof, and that whatever was to be proved, has been.

Here’s a chance for you to discover your own patterns, make conjectures, and create your own proofs.  The source of the patterns will be a calendar.  Follow the example shown below, and then try one for yourself. • If you look at the circled dates above, you will notice all three sums are multiples of 3.
• The conjecture would be, the sum of any three dates in a row will always be a multiple of 3.
• The pattern was found with specific numbers on a specific calendar, but the conjecture is for any 3 dates in a row. This means we need an algebraic way of representing any 3 consecutive dates.
• The variable “x” can stand for any date.  No matter what that date is, the following day’s date will be that date plus 1, or “x + 1”.  Similarly, the third date will be that first date plus 2, or “x + 2”.

Below is an example of how to write a proof for the conjecture on the previous page. Choose any month from any calendar, find a pattern, make a conjecture, and write a proof. Open the document  “calendar proofs.pages”  for creating a calendar proof digitally, or use a copy of the pdf  calendar proofs  for creating a proof on paper. In addition to, or instead of, a calendar, use the document  “mult table proofs.pages”  or  mult table proofs. After finding a pattern, making a conjecture, and creating a proof, go on.

In Chapter 1, we were introduced to function notation, and how inputting values into functions produced unique outputs.

For instance, if  f( x ) = 3x + 2  , then  f( 5 ) = 3( 5 ) + 2  =  15 + 2  =  17  , and only 17.

We also found out, the inputs can be polynomial expressions, which result in polynomial expression outputs as well.

For instance, if  f( x ) = 3x + 2  , then f( x + 1 ) = 3( x + 1 ) + 2  =  3x + 3 + 2  =  3x + 5.It turns out, we can even input bivariate polynomial expressions (polynomials which have two different variables).

Below are snapshots of a program, which randomly generates quadratics in x (a degree 2 polynomial in terms of x).  The challenge is to input a linear polynomial in x and n (a degree 1 polynomial in terms of both x and n), and write the output as a polynomial with the variables in terms of x and the coefficients in terms of n.  Run  program 207  until you can successfully determine f( x + n) for a quadratic function, in terms of x. Use a copy of the pdf  programnotes to neatly record the original function, and all of your work. After finding the output polynomial in terms of x, with polynomials in terms of n for coefficients, go to the next page. It turns out the coefficients in terms of n, in the previous program, have special meanings.

For instance, the coefficient in terms of n, for the x term, is actually a steepness formula for the original quadratic function and its tangent lines. Tangent lines of a parabola (the graph of a quadratic), are lines which touch the parabola at one and only one point. The snapshot to the right, shows the parabola of the polynomial  -3x2 – 9x – 2  (from the previous page),  and a tangent line with a steepness of  -6x – 9,at the tangent point  (-2 , 4). In other words, at the point  ( -2 , 4 ), both the parabola and the tangent line, have a steepness value of  -6( -2 ) – 9  =  12 – 9  =  3. Run the applet  “steepnessformula.html”, to explore more tangent lines of the parabola shown to the right, and to get a sneak peek at some of the mathematical ideas which will be explored later in the book.

The more tangent lines you graph, the closer you get to creating the parabola, or graph, of the original quadratic. To the right are snapshots of a program, which will randomly generate a quadratic polynomial function, determine f( x + n ), use the coefficient of the x term as a steepness formula, and graph enough tangent lines to see the parabola of the original quadratic function.

The parabola formed by the tangent lines, is an example of something called an envelope (The program never directly graphs the parabola). Run  program 208  to see parabolic shaped envelopes being created.There’s a way of using tangent lines to calculate the areas of sections of parabolas, which can help to mathematically show if a model rocket is safe to fly or not when parabolic shaped nose cones or fins are used.  Exploring ways to create parabolas has led to many applications.  Paraboloids (3-D parabolas) are commonly used to help receive and transmit sound, light, and other magnetic radiation. Open the pdf  mirage to see if you can discover why and how paraboloids are used. If you found the picture of the para-”boo”-la at the beginning of the chapter interesting, run the applet  “paraboola2.html”, and  program 214  to create your own. Below is a snapshot of the chapter 2 programs menu program.  Run the program  “2menu.bas”, or its alias to access every program used in chapter 2.

“Mathematics is the gate and key of the sciences…” — Roger Bacon —

“The things of this world cannot be made known without the knowledge of mathematics” — Roger Bacon —

“In any particular theory there is only as much real science as there is mathematics” — Immanuel Kant —

“A book on the new physics, if not purely descriptive of experimental work, must essentially be mathematical.” — P.A.M. Dirac —

“Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.” — Bertrand Russell —