CHAPTER 1

Topic: Terms & Addition

The true test of character is not how much we know how to do, but how we behave when we don't know what to do.

— John Holt

Term

Every topic in Mathematics has a fundamental item that is the cornerstone that needs to be completely understood. The more fully understood the object, the easier the topic. Geometry use figures that can change from triangles to squares to a tetrahedron. We are lucky in Algebra that we only deal with terms. The Art of Algebra is the manipulation of those terms using a solid foundation of Arithmetic. Let us start by answering the question, "What is a term?"

Let us now look at the various parts of a term. There are three parts that we must be concerned with.

Part #1 is the sign of the term. We recall from Arithmetic that we must always be conscious of the sign.

Part #2 is the value. Once again we know the importance in Arithmetic calculations.

The proceeding two parts are going to be merged into a single entity because they really are used exactly like the fundamental item of Arithmetic, integers. This is usually called the Number Part or coefficient.

Part #3 is the letter part. This is the new element we need to deal with. In this example we have a letter part, x3. We have a variable of x raised to the power of 3. It should become obvious that mastery of the exponent rules will be important.

Let us now get a final picture of our, "term."

We can see that it comes down to two parts. All the various computations and techniques that we will employ must always take into account these two elements.

Single Variables

We note that we are only dealing with terms using a single variable. It will not take long until we talk about double variables, triple variables and so on.

Example #1 (Refer to Assignment #1 — Question #1)

Complete the following table.

• The sign is not there so it is positive (+)

• The value is 7

• The number part is +7

• Letter part is x

• Variable is x

• Exponent is not there so it is +1

Example #2 (Refer to Assignment #1 — Question #1)

Complete the following table.

• The sign is negative (-)

• The value is 5

• The number part is -5

• Letter part is t7

• Variable is t

• Exponent is 7

Multi Variables

One of the quirks of the letter part is the fact that it can include more than one variable. Each variable has its own exponent.

Example #3 (Refer to Assignment #1 — Question #1)

Complete the following table.

First we note that our table can handle up to four variables labeled VI, V2, V3 and V4. Consequently, the exponents are designated by El, E2, E3 and E4.

Further to our tip, let us change our term.

• The sign is negative (-)

• The value is 7

• The number part is -5

• Letter part is xy3

• First Variable (V1) is x

• First Exponent (E1) is not there so it is +1

• Second Variable (V2) is y

• Second Exponent (E2) is +3

Example #4 (Refer to Assignment #1 — Question #1)

• The sign is negative (-)

• The value is not there so it is 1

• The number part is -1

• Letter part is a2b21c4

• First Variable (V1) is a

• First Exponent (E1) is +2

• Second Variable (V2) is b

• Second Exponent (E2) is +21

• Third Variable (V3) is c

• Third Exponent (E3) is +4

Like Terms

Let us look at the following definition.

This should be fairly easy to determine.

Example #5 (Refer to Assignment #1 — Question #2)

Place the following in groups of like terms.

5x, 7y, -6xy, 2x2, 4y, yx, -4y

First, let us make sure that all letter parts are in alphabetic order.

5x, 7y, -6xy, 2x2, 4y, xy, -4y

Starting with 5x we note there are no other terms with a letter part of x

So Group #1 is 5x, and we will cross it out to focus on the rest.

[MATHEMATICAL EXPRESSION OMITTED]

Starting with 7y we can see two other terms with the same letter part "j".

[MATHEMATICAL EXPRESSION OMITTED]

So Group #2 is 7y,4y,-4y and we will cross them out to focus on the rest.

[MATHEMATICAL EXPRESSION OMITTED]

Starting with -6xy we can see one other term with the same letter part "xy".

[MATHEMATICAL EXPRESSION OMITTED]

So Group #3 is -6xy,xy and we will cross them out to focus on the rest.

[MATHEMATICAL EXPRESSION OMITTED]

This leaves us with Group #4, 2x2.

In summary,

Example #6 (Refer to Assignment #1 — Question #2)

Place the following in groups of like terms.

-4y, 8x2y, 9xy3, 6y, -7x, 2x, -3y, 11, 7x2, 2y, -11x2y2, 5x2y, 3x2y2, 4y, -9x, 7xy3

First we note they are all, "alphabetically correct."

Addition of Monomials

Before we get into the techniques of addition, let us get a handle on the expression, "monomial." In the world of Algebra, we frequently see expressions with more than one term. First, we need to establish that terms are separated by plus or minus signs once there are no brackets left. In addition, all like terms must be simplified. Before we get ahead of ourselves let us look at expressions involving more than one fundamental unit.

Let us look at the case in Integers.

The expression:

-2 + 5 - 8 + 9 - 2 - 3 + 5 - 4 - 6

may appear to be a long list of additions and subtractions, BUT we know that is NOT the case.

This expression can be written as:

(-2) + (+5) + (-8) + (+9) + (-2) + (-3) + (+5) + (-4) + (-6)

This is an example of 9 integers, some positive, some negative, being added together.

Moving to algebra, we can look at expressions in Algebra.

3x - 7y + 6x2y3

We can apply the same principles and rewrite this expression as follows.

(+3x) + (-7y) + (+6x2y3)

We note that we used an assumption to place the plus sign in the first bracket. This is an example of somebody adding three terms.

At this point, we can define algebraic expressions by the number of terms.

Definitions

First, the terminology monomials, binomials and trinomials are used, whereas the terms quadranomial, pentanomial and hexanomial are more figments of my imagination.

By way of explanation, any expression with one or more terms is referred to as a polynomial, because, "poly," means, "any number." Hence, every binomial is a polynomial, but not every polynomial is a binomial.

Now to start our explanation of Addition in Algebra, we will first focus on the simplest of terms, the monomials.

Consider

5x + 7x

We know this can be written as

(+5x) + (+7x)

I will now revert to the oldest trick in the bag of scams that Algebra Teachers have used since the beginning of time.

"If I asked you how much is 5 APPLES plus 7 APPLES ? Most of you would reply, hopefully, 12 APPLES."

Let us look at that a little more carefully.

We can say

(+5x) + (+7x) = 12x

5 APPLES + 7 APPLES = 12 APPLES.

Consider something a little different

5x + 7y

We know this can be written as

(+5x) + (+7y)

Using the oldest trick in the bag of scams that Algebra Teachers have used since the beginning of time.

"If I asked you how much is 5 APPLES plus 7 ORANGES? Most of you would reply, hopefully, you cannot add apples and oranges. (PLEASE, PLEASE do not say fruit!!)"

Let us summarize these observations.

Example #7 Simplify the following

9y - 11y

First, they are like terms !

Number part 9 - 11 = -2

Letter part is x

= -2x

Example #8 Simplify the following

4y + 11t

First, they are NOT like terms !

THERE IS NO SIMPLIFICATION

[therefore] = 4y + 11t

Example #9 Simplify the following

-3x - 5x

First, they are like terms!

= -8x

Addition of Polynomials

This section will appear a little awkward, since we are moving ahead with addition before we have done multiplication. To an Algebra purist this may make these examples a little uncomfortable, but we will get there.

Example #10 (Refer to Assignment #1 — Question #4)

Add the Algebraic Expression

3x2 - 7x + 2

To

5x2 + 8x

We are, in effect, adding a trinomial to a binomial.

3x2 - 7x + 2 + 5x2 + 8x

Once again, the purist will want brackets. We can rewrite this, using brakets:

(+3x2) + (-7x) + (+2) + (+5x2) + (+8x)

Now we know the only type of monomial addition we can do must have like terms.

[MATHEMATICAL EXPRESSION OMITTED]