Algebra in Words: A Guide of Hints, Strategies and Simple Explanations (2014)

REVIEW OF THE BASICS

First, we must look at a few basics that will be used and referred to throughout this book. From year to year and class to class, you may have grasped the majority of the material you’ve learned, and built a good foundation. But, say, over a summer or holiday break, or just from not using it enough, you may have forgotten a few of the more obscure principles. These are here to quickly bring you back up to speed.

The Real Order of Operations: GEMA

You may remember that when adding terms, you will get the same sum regardless of the order in which you add. You may also remember that the same applies to multiplication: You will get the same product regardless of the order of the factors you multiply. These are “commutative properties.” However, this does not apply to subtraction or division.

For subtraction and division, order matters. Order matters, including when subtraction and/or division is mixed in with terms being added and/or multiplied. Since order matters, there are a set of rules in place to help us calculate numbers and terms in the proper order, and to put consistency into the way we do math. These are the order of operations. Often, books or instructors do not teach this completely correctly. Here, it will be explained completely, with nothing left to be misinterpreted.

1. Simplify inside Groups first, if possible, from inner to outer. A group is a set of (parentheses), [brackets], {braces}, overall numerators, overall denominators, and radicands.

2. Exponents or roots, whichever comes first, from left to right.

3. Multiplication or division, whichever comes first, from left to right.

4. Addition or subtraction, whichever comes first, from left to right.

The Truth about PEMDAS

Many students are taught the acronym and mnemonic device PEMDAS, which stands for “Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.” I must warn to you be careful of PEMDAS; it is misleading and incomplete. If you learn to rely on it, it can fail you. Letter by letter, here’s why:

P: The first order is groups, of all sorts, as described above. If you think of parentheses only, and you get to other groups, you might think it applies to parentheses only. A more appropriate first letter and word should be G for Groups. Groups include, but are not limited to parentheses.

E: This makes you think of exponents only instead of roots as well. This isn’t a big deal, since radicals can be converted into exponent form (when they are, they are called rational exponents), but they are often in root or radical form, so you must be prepared for that. When both exponents and roots appear in an equation, do whichever comes first, from left to right. Also, remember that radicands should be simplified first, if necessary, as they are technically a group, as mentioned in the first step.

MD: The reason this is misleading is because some people interpret this to be chronologically literal. In other words, some see M before D and think multiplication must happen before division, but in fact, it means that any Multiplication or Division come before any Addition or Subtraction. But multiplication or division should be treated with the same priority, and you’re supposed to perform whichever of the two operations comes first from left to right, in the direction you read. For instance, if, in an equation, a division sign comes before a multiplication sign, from left to right, you divide first and multiply next.

AS: Many often interpret this as “A is before S,” (as they think M comes before D) but in fact they are of equal priority in the way M & D are. Addition or subtraction should only be performed after all other operations are completed. Then you perform either addition or subtraction, whichever comes first from left to right. If subtraction comes before addition, you would do the subtraction first, then the addition next.

If anything, consider PEMDAS a loose reminder of the complete Order of Operations, although if it were up to me, PEMDAS be thrown away completely and replaced with GEMA:

1. Groups (simplify, inner to outer)

2. Exponents or roots

3. Multiplication or division

4. Addition or subtraction

Actually, since:

·   Roots are technically a form of exponents, when converted to rational exponents^*;

·   Division is technically multiplication of a fraction; and

·   Subtraction is technically addition of negative numbers,

it could even be condensed to just:

1. Groups,

2. Exponents,

3. Multiplication,

4. Addition

*Note: Roots are converted to “rational exponents” when the radical sign is removed and the root-number is moved to the denominator of the exponent.

The Unwritten 1

You must remember that “1” is often not (required to be) written or shown, but is still there. I give an example showing the “1” written as both the coefficient in front of x (could be any variable), the denominator of x, the power of x, and the denominator of the power 1 (of x). Moving right, I show it with the denominators removed, and then all the 1s removed, showing just x.

However fundamental this may seem, it is a concept students often question or forget, and for that matter, sometimes fail to implement when necessary. Here are reasons it is helpful to remember that “1” is still there:

·   As a coefficient so its associated variable can be added to other like-terms, such as:  x + 3x = 1x + 3x = 4x

or as in adding radicals:

·   As a denominator, especially for (fraction conversions to like-fractions during) addition/subtraction of fractions as in:

·   As a denominator for (inverting, then multiplying a fraction during) division of fractions as in:

·   As a power or root, especially for multiplying factors (of a common base) with exponents (in which you add the exponents), as in:

·   As a power or root, especially for dividing factors (of a common base) with exponents (in which you subtract the exponents), as when simplifying:

·   As a valid placeholder after a Greatest Common Factor has been factored out, as in:  3x + 3 = 3(x + 1)

It is also worth reminding you about the unwritten one associated with a negative sign. Take a look at the following examples: -4, -x, - . These can be thought of as:

(-1)(4),

(-1)(x), and

(-1)(), respectively.

Property Crises of Zeros, Ones & Negatives

There are many fundamental properties involving various operations with 0, 1 and negative numbers (I will focus mostly on “-1”). Some are easy to remember, however, some are easy to confuse or forget, but they are vital to get right. In textbooks, these are often thrown at you from different directions, at different times, often with vocabulary or definition-like labels. These are properties involving multiplication, division, exponents and roots. For ease and convenience, I’ve summarized the important ones here in this section, leaving out the labels, but showing the property, then explaining it in words, the way you might say it, hear it, or hear it in your head. Hearing (or reading how you might hear) these should drive home an extra dimension into your brain, so you can more easily recall them later.

(#)(1) = #.

In words: Any number times one equals itself, always, with no exceptions.

# ÷ 1 = #,

also seen as a fraction: = #.

In words: Any number divided by one equals itself, always, with no exceptions. For fractions: Any number over one equals itself (the top number). Likewise, any number can be assumed to be over one, and can easily be converted to a fraction by putting it over one.

Any # or term ÷ itself = 1,

also seen as a fraction: .

In words: Any number or term divided by itself equals one, except when that number is zero. For fractions: Any number (or term) over itself equals one, except when those numbers are zero. Another way to say it is, “any non-zero number over itself equals one.” See the next example.

# ÷ 0 = undefined,

also seen as a fraction: = undefined.

In words: Any number divided by zero is undefined. Or as a fraction, any number over zero is undefined. Any number divided by zero does not equal zero. You might say “you can’t divide any number by zero.”

1 ÷ 0 = undefined,  also seen as a fraction: = undefined.    

In words: One divided by zero is undefined, because any number divided by zero is undefined, as shown in the previous example.

0 ÷ any # = 0, with the exception of when the denominator is 0;

Also seen as the fraction: = 0, except when the denominator is zero.

In words: Zero divided by any number equals zero, with one exception. The exception is: Zero divided by zero is undefined, because any number divided by zero is undefined. Another way to think of it is:

0 ÷ any non-zero # = 0, and  

= 0.

0 ÷ 1 = 0,

also seen as a fraction: = 0.  

In words: Zero divided by one equals zero. In fraction form: zero over one equals zero.

This exemplifies two other properties, previously shown. This example follows that:

·   Any number over one equals that number (the top number), and

·   Zero over any non-zero number equals zero.

0 ÷ 0 = undefined,

also seen as a fraction: = undefined.    

In words: Zero divided by zero is undefined, because any number divided by zero is undefined.

·   This is the exception to the rule that “any number over itself equals one.”

·   It is also the exception to the rule that “zero divided by any number equals zero.”

·   Zero divided by zero does not equal zero, as some mistakenly think.

This property is especially useful when looking at:

·   Slopes of Vertical Lines (see: A Vertical Line, and: When x₁ = x₂), and

·   Extraneous Solutions (see: Solving Equations with Rational Expressions and Extraneous Solutions).

So far, we have looked at many properties and examples which result as: “Undefined”. When doing these functions on a calculator, you might get “error.” For a more on that, see: What Does “Error” Mean?

All but one of the properties shown so far involve division with 0 and 1. The following section will focus on exponents and roots involving 0, 1 and negative numbers.

Base¹ = Base  

In words: Any base to the power of 1 = the base …which is the same as to say:  

Any#¹ = itself.

In words: Any number (base) to the power of one equals itself (that base), as in the next example:

1^{Any #} = 1.

In words: One to the power of any number equals one, with no exceptions.

0¹ = 0.

In words: Zero to the power of one equals zero.

1⁰ = 1.

In words: One to the power or zero equals one.

Any#⁰ = 1.

In words: Any number to the power of zero equals one (with one exception; see next example). Another way to remember it is:

Any non-zero#⁰ = 1.

In words: Any non-zero number to the power of zero equals one.

0⁰ = undefined (and does not = 1 or 0).

In words: Zero to the power of zero is undefined. Likewise, zero to the power of zero does not equal one or zero.

It is important to remember the names of the components of a radical. The term inside the radical is the radicand. The term in the “v” is the root. Also, although the symbol, shape and setup of a radical may closely resemble long division, they are not the same in any way.

In words: The square root of one is one.

Note: The square root means “to the root of two,” but the “2” is commonly unwritten. When the radical sign has no number written in the “v” area, it is implied to be two, meaning the square root.

In words: Any non-zero number root of positive one equals one.

.

In words: Any radicand to the root of zero is undefined. The “root of zero” can also be called the “zeroth root” or the “zeroeth root.” Also, sometimes this answer is given as “infinity (¥),” instead of undefined.

In words: The square root of zero equals zero.

In words: Any non-zero number root of zero equals zero.

no real solution… and may be expressed as “i".

In words: The square root of negative one has no real solution. It is said to have no “real” solution, because the symbol (letter) “i” (for “imaginary”… as opposed to “real”) can be used interchangeably with . In that way, you can consider as having a solution, but since it is not a real number, it is said to be “no real solution.” The symbol i is useful to manage multiple occurrences of the square root of negative one within an expression or equation.

The following examples demonstrate the importance of the second order of operations, as well as the complete and proper wording of that rule, mainly that roots are to be computed first before any other multiplication and division or addition and subtraction. Notice specifically that the root is (to be) computed first, and any factors or negative signs outside the radical are applied next.

In words: Negative of the square root of positive one equals negative one. It is important to notice that the negative sign is outside the radical and is thus attributed to the result of the radical, which is calculated first, according to order of operations.

no real solution

In words: Any even number root of negative one has no real solution… and may be expressed in terms of “i".

= no real solution… and may be expressed as “-i".

In words: The negative of the square root of negative one has no real solution, and thus might be answered as “no real solution,” or as “negative i.” It is important not to mistakenly see this as “a negative times a negative equals a positive,” because the even root of any negative number has no real solution, first and foremost, regardless of if it is multiplied by a negative outside the radical.

Remembering that the square root (or any even root) of any negative number can’t be found (yields no real solution) is important to remember when:

·   solving quadratic equations,

·   using the quadratic formula, and

·   understanding the graphs of parabolas.

In words: The cube root of negative one equals negative one.

In words: An odd number root of negative one equals negative one, as in the last example.

In words: The negative of the square root of positive one equals negative one. This follows the order of operations by taking the root first (of positive one), then attributing the negative sign from in front of the radical next, which is really multiplying (the result of the radical, which here, is 1) by negative one (the “1” outside the radical associated with the negative sign is unwritten).

The following examples involving exponents also demonstrate the importance of order of operations, and the prioritization of groups (here, parentheses) and exponents, which precede multiplication (or division and addition or subtraction). The placement of a negative sign with regards to the parentheses and the base number is very important.

(-1)² = 1

In words: The base negative one squared equals positive one. For this to be true, the negative must be associated to the base: one inside the parentheses.

(-1)^{even #} = 1

In words: Negative one to the power of any even number equals positive one.

(-1)³ = -1

In words: The base negative one cubed equals negative one.

(-1)^{odd #}  = -1  

In words: The base negative one to the power of any odd number equals negative one.

-(1)² = -1

In words: A negative outside the parentheses times positive one squared equals negative one. Order of operations dictates that exponents must be performed first, and positive one squared equals positive one. Since the negative is outside the parentheses, it’s like multiplying negative one times positive one (positive one being the result of one squared).

Finally…

-1² = -1

In words: Negative one squared equals negative one. Order of operations plays a prominent role in this answer. Specifically, the exponent must be applied to the base of one first, and the negative is applied next. This example is often mistakenly answered “positive one,” because some mistakenly see this as the square of negative one, or negative one times negative one. This would be different if the negative were inside the parentheses with the one, and the power of two was outside the parentheses, as shown four and five examples back.

This example not only shows the importance of the placement of signs and parentheses (or lack of), but it also shows the importance of how the math is heard or spoken. If you look back at how this was translated in words, you can’t assume that the negative is in parentheses with the base “1”. For that reason, when speaking an equation containing parentheses, you should be specific as to where they start, end, and what is inside them.

I started you off with some fundamental properties. This review will help you brush up on the little things many people commonly forget from their previous math experience. Having these reminders will give you an extra edge in solving problems (especially problems that appear to be complicated, but aren’t, once you apply these equalities).

For more on radicals and commonly used roots, see the: Radicals, Roots & Powers section later in the book.

Integers & Whole Numbers

The words “integers” and “whole numbers” are commonly confused and misused among students. These words are used to properly communicate math, so it is important you use them properly, as well as understand when they are used.

Whole numbers are positive, non-decimal, non-fraction numbers. The main aspect students often forget about whole numbers is that they can only be positive. They can also be defined as “positive integers.” One place you will see whole numbers used are in chemical formulas and chemical reaction equations. The subscripts in a chemical formula can only be whole numbers. And the coefficients in a balanced chemical reaction equation also can only be whole numbers.

Integers are also non-decimal, non-fraction numbers, but can be positive and negative. Any and all whole numbers are also integers (whole numbers are the positive integers).

The word “integers” is often seen and used during factoring of trinomials and quadratics. When factoring using the Trial & Error/Reverse FOIL Method, you are told to look at the “integer factors of the first (ax²) and last (c) terms” (although, sometimes fractions and decimals are permitted too, where applicable, which is generally when the first (ax²) and last (c) terms are fractions or decimals to begin with).

Also, the absolute value of any integer results in a whole number.

Prime Numbers

A prime number is a number which is not divisible by any number other than itself and 1 (without resulting in a non-integer number). Meaning, if you divide a prime number by anything other than itself or 1, the result will be a decimal (or equivalent fraction). The first thirty prime numbers are listed here as a reference, so if you ever need a place to quickly check a number, you can refer here. You can easily find more on the internet.

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109…

Why are these important to know? Because knowing whether a number is prime tells you whether you can proceed to factor it. If it is not prime, you can factor it. If it is prime, you cannot factor it. This is useful to know when you are factoring numbers (sometimes using a factor tree; see your textbook for more on factor trees); when doing factor trees, your goal is to factor all numbers into prime numbers. This may be used when finding an LCD or in the process of factoring trinomials into binomials. You also use factoring with radicals, but in those cases, you don’t always need to factor to prime numbers. For more on that, see: Manipulating & Simplifying Radicals.

Polynomials can also be prime, which in their case means they can’t be factored into polynomials any smaller than themselves. For an example of a prime polynomial, see: The Sum of Two Squares.

It is also worth noting that the opposite of a prime number is a composite number, which is a number with factors other than itself and 1.

Is 51 a Prime Number?

The number 51 is not a prime number, but is often mistaken for being prime. I guess it just somehow looks prime… whatever that means. It’s clearly not even (so it’s not divisible by 2), and it’s just a number we don’t use as regularly as the numbers 0 through 50. Plus, many other prime numbers end in 1, as seen in the list of Prime Numbers. But don’t let this number slip under your radar when you need to know whether it is prime.

How can you tell it isn’t prime? Because it follows the technique that: if you add the digits of a number, and that sum is a multiple of 3, then that (original) number is also divisible by 3 (and another integer). The digits 5 + 1 = 6, and 6 is clearly a multiple of 3, therefore 51 is also divisible by 3. The number 51 can be factored to (3)(17), which is actually the prime-factorization of 51.

These factoring strategies are discussed more in The Procedure for Prime Factoring. Also, you will notice the number 51 in The Prime Factor Multiples Table.

What is a Term?

A term is any number or variable or combination of them that either stands alone or is set apart by other terms with a “+” or “-“ sign. Terms may be made up of factors, but terms are not factors themselves. They can’t be factors because factors are multiplied, not split apart by plus or minus signs. Be careful not to use “factor” and “term” interchangeably.

What is a “Like-Term”?

A like-term is a term with a common base and common exponent. Terms must have both the same variable (letter) and the same exponent to be eligible to be combined. Only like-terms can be added or subtracted (*see note below). If there are terms that are not variables, they must only be constants (numbers) which count as like-terms and can simply be added or subtracted together.

“Like-terms” also apply to addition & subtraction of radicals. Since any radical or root can be converted to and expressed as an exponent (when it is, it is called a rational exponent), this follows the definition of “like-terms.” “Like-radicals” can still be added and subtracted even if they’re not in rational exponent form.

You may not deal with radicals when you begin the basics of algebra; you usually learn about these later. For more on how like-terms apply to radicals, see: Manipulating & Simplifying Radicals.

*Note: This does not mean like-terms cannot be multiplied or divided, because they can be. The statement referred to above is meant to accentuate that non-like-terms cannot be added or subtracted. To be clear, any terms can be multiplied and divided; they don’t have to be “like.” Non-like-terms can be multiplied (however, when they are multiplied, they are technically factors). Consider the terms: 3x² and 4x². They are like and can be multiplied to get 12x⁴.

What is a Factor?

A factor is a number or variable that is or can be multiplied by another number or variable. Factors combine via multiplication to make a term (and yes, factors are multiplied to give a product, but this section is meant to help distinguish between factors and terms, as they are often used incorrectly interchageably). But often times, to serve their function, factors are not multiplied together, rather they are factored from larger numbers or terms and shown as unmultiplied, individual factors. Unmultiplied factors just look like numbers or variables standing next to each other (often each in parentheses). Converting a larger number or term into factors is done by factoring.

Factoring

Factoring is a way of breaking down a larger number or term into its factors. Factoring is performed using division and trial & error (explained in The Procedure for Prime Factoring). This is often done to compare factors to other factors, so when common factors are found, they can be cancelled out or grouped together by adjusting the exponent, depending on the situation. Also, terms are often factored into prime factors when you do a factor tree. For more on factoring numbers, see: Prime Numbers and The Prime Numbers Multiple Table.

Factoring terms with variables is a bit different. It is explained in The Greatest Common Factor, a few sections later.

The Procedure for Prime Factoring

For smaller numbers, finding two starting factors may be easy to figure out. But for a large number, it may not be as easy, and this is a place student run into trouble. In this case, you can use a process which starts from an easy place, but you must know the process and follow it properly and sequentially. This process involves looking for a small prime number that the number you are factoring is divisible by (actually, you are looking for the smallest prime number that the number you are factoring is divisible by).

This is the process: Ask yourself if the number you are factoring is divisible first by the smallest prime number (2), then the next larger (3), and keep working your way up until you find it. You may need to check back to the list ofPrime Numbers once you go beyond the prime numbers you know by heart. Within this process are sub-processes that you should apply as you work your way up the prime numbers. The following are those helpful sub-processes:

For 2: Check to see if the number you are factoring is even. Specifically, check to see if the last digit of the number is even.

  If it is even, then the original number you are factoring is divisible by 2. Then you should divide it by 2 to find the other factor.

  If it is not even, then the number is not divisible by 2, nor is it divisible by any other even number: 4, 8, 10, 12, etc. Next, test if it is divisible by 3…

For 3: Add up the digits in the number you are factoring. If the sum of the digits is divisible by 3, then the original number you are factoring is also divisible by 3. Then you should divide it by 3 to find the other factor.

  If the sum of the digits is not divisible by 3, then the original number is also not divisible by any other multiple of 3 such as 6, 9, 12, 15, 18, etc. Next, test if it is divisible by 5…

For 5: If the number you are factoring ends in either 5 or 0, then it is divisible by 5.

  If doesn’t end in 5 or 0, then it is also not divisible by any other multiple of 5. Next, test if it is divisible by 7…

For 7 and higher: Take the number you are attempting to prime factor, divide it by 7, and analyze the quotient. If the quotient comes out as a whole number, then the prime number 7 you divided by is a factor, and the quotient is the other factor.

If the quotient comes out as a decimal, then the number you divided is not a factor. Repeat this process by attempting to divide by the next higher prime number until you get a quotient which is a whole number. You may need to repeat this process on the whole number quotient until you end up with all prime numbers (factors).

You might also refer to my Table of Prime Number Multiples to expedite the process.

Once you have found all the factors needed, or have completed the factor tree, present your factors according to your book or teacher’s instructions.

The Prime Number Multiples Table

When students learn multiplication, they are sometimes given a table showing the numbers 1-10 (or higher) along the top (row) and down along the left (column), to easily find the product of two numbers. Students are also taught prime numbers, especially when learning to (prime) factor numbers. As in the example of “51” discussed previously, there tend to be some higher numbers which are not prime numbers, but may either appear to be, or students give up trying to figure out if they are through dividing up the prime numbers list, according to the suggested procedure for factoring large numbers (and finding what they are divided by). So I created a helpful tool I call a Prime Number Multiples Table. It is important you understand what it is and how to read it. This table is meant to be an extension for the procedure used to prime factor a number, according to The Procedure for Prime Factoring.

First, you will notice that the numbers across the top row and down the left column are prime numbers only. Also, notice that the prime numbers 2 and 5 are not shown in the table. That is because it would be a waste of space because you can easily tell if any number, no matter how big, is divisible by 2 (because it would be even) or 5 (because it would end in 0 or 5). Even though the method for determining if a number is divisible by 3 is simple, I included 3 anyway, because you can’t always tell if a number is a multiple of 3 just by looking at it, the way you can for an even number.

If and when you are attempting to factor a large number, you should look for that number in this table. If you find it, then you automatically know it is a product of the two prime numbers from the top row and left column it comes from.

If your number is:

·   less than 1173,

·   not found in this table,

·   (and not a multiple of 2, 3 or 5, as you should have checked in the beginning),

then your number must be prime. If your number is greater than 1173, then you must continue checking it (by dividing it) with prime numbers beyond the prime number 53.

In short, this table will save you a lot of guess-work and trial & error. Although this is meant to be a time-saver and a tool to help you familiarize yourself with products of prime numbers, you still must be able to figure it out the long way (without the table), as I’m sure you won’t be permitted to use this table or book during a test.

The Greatest Common Factor (GCF)

The GCF is the largest factor that can be factored out of every term involved. Another way to think of it is: the biggest factor that each term can be divided by, without resulting in a fraction or decimal. The GCF is found and used in the overall simplification process for:

·   Reducing fractions, and/or

·   Factoring a series of terms,

More specifically, the GCF is mainly found and used for two reasons:

1.  To cancel common factors in a fraction to reduce that fraction. In this case, you would find the greatest factor that is common in the numerator and denominator and proceed to cancel them out (to 1).

2. To extract the GCF out of a series of terms, which, once you find it, you factor it out of each term, and the GCF goes outside (usually to the left of) a set of parentheses.

The Least Common Denominator (LCD)

The Least Common Denominator (LCD) is also known or used in some contexts as the Least Common Multiple (LCM). Actually, all LCDs are LCMs, but LCD is just specific to denominators. Books usually introduce the concept and procedure for finding the LCM in preparation for learning LCDs. In this book, I will refer solely to the LCD.

The LCD is commonly found and used for three main reasons:

1.  To convert fractions into “like fractions” for adding and subtracting fractions (including: rational expressions), both of which are discussed later in the book.

2. To eliminate all denominators (and thereby all fractions) in an equation by multiplying each fraction by the LCD. In equations with fractions, this sometimes must be done in order to solve.

3. To reduce Complex Rational Expressions.

When doing problems involving an LCD, I recommend you write “LCD = (then show the factors here)“ on your paper and fill in the factors as you gather them. This gives you a place of reference to keep track of your factors while you look back at the fractions in the problem. Sometimes you should use factor trees to help you find the LCD, especially if the LCD isn’t obvious to you. Actually, using factor trees is a good habit. I often find that students want to avoid doing factor trees because they think the LCD is more obvious than it often is, but this is a major mistake. You have a higher chance of finding the correct LCD by doing factor trees. For more how to find the LCD and factor trees, please refer to your textbook, as I do not give the procedures in this book. But also, see: The Procedure for Prime Factoring.

GCF vs. LCD

Students commonly confuse the meanings and applications of the GCF and LCD, probably because they both involve factors and both are used for similar reasons. Both are used for simplification purposes and both can be applied in some way to fractions. In this section, I will go through a few brief hints to help you differentiate between the two.

First, you must understand their general uses, as you can read in the previous section. Next, let’s break down the words and pay attention to some common, associated key words.

The best key words to associate with GCFs are: “out,” “smaller,” “found” and “division.”

·   GCFs are always factored out of a term or a number of terms, to make those terms smaller.

·   When used with fractions, GCFs are factored out of the numerator and denominator to make fractions smaller. GCFs are factored out by dividing terms by the GCF.

·   A GCF can only be found; it is not made. It is a factor that is already there, or already within the terms of interest. If there is no GCF present, then the fraction is already completely reduced, or the terms are already completely simplified.

·   Whereas an LCD may already be there (if the largest denominator is already the LCD), but it can also be made…

The best key words to associate with LCDs are: “bigger,” and sometimes “multiply,” and “made.” The starting point is looking at the largest denominator. The LCD used may be the largest of the original denominators (you would have to evaluate to make that realization). But if it is not, then:

·   The LCD will ultimately be bigger than all the original denominators.

·   The LCD will made by multiplying the appropriate factors.

·   The fractions involved, which are to be converted (if necessary), will have their numerators and denominators multiplied to make fractions with bigger numbers than before.

These key words can seem a bit mixed when in reference to simplifying complex rational expressions. In this case, the LCD of all mini-fractions is either already there, or made (by multiplication), then multiplied by all mini-fractions involved. This ultimately reduces the complex rational expression, but along the way, some or all of the numerators of the mini-fractions become bigger.

Lastly, it is important how you use the word “find.” In the GCF context, “find it”, means “look for it.” In the LCD sense, you look to see if the largest denominator is already the LCD, otherwise, you have to find (meaning make) it.