RATIONAL EXPRESSIONS - A Guide of Hints, Strategies and Simple Explanations - Algebra in Words

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

RATIONAL EXPRESSIONS

By its technical definition, a rational expression is a fraction that contains polynomials. But since, to be a polynomial, it must contain at least one variable (otherwise, if it just contained constants or numbers, it would be considered a regular fraction), my definition is: A rational expression is a fraction containing (one or more) variables. Notice the root word “ratio.”

Sometimes, you will be asked to simplify a rational expression and sometimes you will have to solve an equation containing rational expressions. When dealing with rational expressions, you must know how to properly simplify them.

Procedure for Simplifying Rational Expressions

It is important and helpful that you can clearly see each numerator and denominator as separate pieces because they will need to be simplified individually, first, before continuing.

0. For this reason, I recommend putting parentheses around numerators and denominators. (This reinforces the first rule of Order of Operations, being that numerators and denominators are in fact “groups” but are rarely written with parentheses. Putting parentheses around them makes them look more like groups, and will remind you to treat them as such).

1. Factor each numerator and denominator separately, completely factor the polynomials.

2. Look for factors that are alike in the numerator and denominator, then cancel them out to 1 (correctly, by avoiding: The Wrong Way to Simplify a Rational Expression).

Procedure for Adding & Subtracting Rational Expressions

This follows the procedure for adding & subtracting fractions, only now, variables are involved. You add & subtract rational expressions the same way as fractions by finding and using the LCD. Remember, the goal for adding & subtracting rational expressions is to properly combine them so that one rational expression remains. Follow this procedure (the first steps, 0a – 0c, are more preparatory steps):

0a. Put parentheses around all numerators & denominators.

0b. Look for any minus signs in front of a fraction. If there is a minus sign in front of a fraction, you must distribute it through its associated numerator. To do this, replace the minus with a plus and change every sign in its associated numerator. This is often an overlooked step which results in +/- sign errors later on.

*Note: A big error students make here is applying the negative sign to only the first term in the numerator, instead of applying (distributing) it to every factor in the numerator.

0c. Put all terms in numerators & denominators in descending order.

1. Factor all the numerators & denominators separately.

2. Look at all factors of the denominators and determine the LCD.

·   Write the LCD off to the side so you can refer to it, and write the LCD in the denominator to the right of the = sign (as of now, the numerator is blank; you will fill in the numerator in step 5). Leave the LCD asun-multiplied factors (which will make simplifying easier in the later steps).

3. Look at each denominator (of each fraction) and determine what factors are missing (I call these the “missing-factors”) if any, to complete the LCD in each.

4. Multiply the numerator & denominator of each fraction by its missing-factor(s)

5. Multiply & distribute factors in the numerator and write the products in the numerator above the LCD written on the right side of the = sign. Don’t forget to transfer the proper signs.

6. Simplify the terms in the new numerator.

6a. Combine like-terms.

6b. Look for a GCF, and whether there is one or not, try to factor completely.

7. Simplify: Cancel out any common factors in the numerator & denominator.

·   Make sure you avoid the wrong way to simplify a rational expression.

Simplifying a Complex Rational Expression

Simplifying a complex rational expression is something you will surely be required to do on a final exam.

A complex fraction simply means fractions within a fraction. Since rational expressions are fractions containing variables, a complex rational expression means rational expressions within rational expressions… or simply:

fractions-containing-variables within fractions-containing-variables.

There are two ways to simplify complex rational expressions:

1. The “All-LCD Method.” Multiply all mini-fractions by the LCD of all mini-fractions.

Or

2. Simplify the overall numerator & overall denominator first (by applying the rules for addition & subtraction of fractions), separately, into one fraction each, then divide the top-fraction by the bottom-fraction (see: Dividing Fractions).

Either method, if used correctly, will yield the same result.

When should you use each method?

Method 1, the “All-LCD-Method” avoids needing to add & subtract fractions and you only need to find one LCD. This is typically easier and more preferred. When in doubt, default to using this method. Although this method is more straight forward, it is also tedious and may make your paper messy. For that reason, mistakes are often made in the process for not being able to read your own writing, writing too small, or not leaving enough room. When doing these problems by this method, be sure to leave plenty of room on the paper and write clearly.

Method 2, “Simplifying the Overall Numerator and Overall Denominator Separately” is typically used when the addition & subtraction of fractions in the overall numerator and overall denominator will be a quick and easy procedure. This method is also usually selected when the variables in the overall numerator are distinctly different than those in the overall denominator. The reason for this is because finding and using the All-LCD-Method may introduce variables into opposite parts of the fraction that will require extra and tedious steps (such as factoring) to get to the end.

Ultimately, the choice of method is more dependent on the student’s preference, as both methods are tedious with a number of intermediate steps, but will still yield the same outcome. Because of how tedious they are, you may choose to stick with the one you gravitate towards and work on getting good at it. The two procedures will be given next, at first in a very detailed form, and then in a condensed form, so you can refer back to either version. These may help you decide on your preference.

All-LCD Method (detailed version):

0a. If there are whole numbers or polynomials that are not fractions, it’s a good idea to put them over 1, to make them fractions.

0b. I recommend putting all polynomial numerators & denominators in parentheses. Books often leave them without parentheses, but using them makes it easier to view and use the polynomials or factors to continue.

1. Completely factor all numerators and denominators, if possible.

2. Find the LCD of all the mini-fractions involved.

3. Multiply the LCD by the numerators of all mini-fractions. (See Note 4).

4. Simplify all mini-fractions by applying the rules of multiplication & division of bases with exponents. In this step, all denominators of mini-fractions should cancel out. The factors of the LCD are intended to directly cancel out (with) every entire denominator of all mini-fractions. However, this may still leave the remaining, un-cancelled factors from the LCD, if any, in the numerators of the mini-fractions, and this is expected.

The rational expression is now simple, not complex, as there is now only one numerator and one denominator.

4a. Simplify the (new) numerator. Multiply, distribute and combine like-terms, where possible.

4b. Simplify the (new) denominator. Multiply, distribute and combine like-terms, where possible.

  4c. Arrange all terms into descending order.

5. Completely factor the numerator and denominator, separately. That means factoring out a GCF first, if there is one, and/or factoring the polynomial, if possible, (into smaller polynomials).

6. Cancel out any common factors in the numerator & denominator.

*Note 1: Sometimes factors will cancel out in the last step and sometimes none will. Be ready for either scenario.

**Note 2: At this point, it is up to your professor if he/she wants you to multiply the factors in the numerator & denominator (individually) for your final reported answer. I personally prefer them to be left in factored form.

***Note 3: Don’t commit the frequently made mistake: The Wrong Way to Simplify a Rational Expression.

Simplify Overall Numerator & Overall Denominator Separately Method (detailed version)

0a. If there are whole numbers or polynomials that are not fractions, set them over 1, to make them fractions.

0b. I recommend putting all polynomial numerators & denominators in parentheses. This makes it easier to view & use these polynomials or factors to continue.

1. Completely factor all numerators and denominators, if possible.

2. Next, you will be simplifying the overall numerator & overall denominator separately, using their own LCDs.

2a. Find the LCD of the fractions in the overall-numerator

2b. Find the LCD of the fractions in the overall-denominator.

3. In the overall numerator and denominator, separately, use the procedure for adding & subtracting rational expressions.

At this point, you still have a complex rational expression, but now with only one (unsimplified) mini-fraction in each the overall-numerator and overall-denominator.

4a. Leave all denominators of mini-fractions as un-multiplied factors.

4b. Simplify all numerators of mini-fractions by combining like-terms.

4c. Factor the numerators of the two mini-fractions, separately.

By this point, there should still be one fraction each in the overall-numerator and the overall-denominator, but now, each mini-fraction is simplified. You are now ready to…

5. Divide the top mini-fraction by the bottom mini-fraction using the rule for dividing fractions.

You now have a (simple) rational expression (one numerator and one denominator). They may or may not already be simplified.

6. Simplify: Cancel out common factors from the numerator & denominator.

All-LCD Method (short version)

0. Put whole factors over 1 and put all overall numerators & denominators in parentheses.

1. Factor all numerators & denominators.

2. Find the LCD of all mini-fractions.

3. Multiply all mini-fractions by the LCD.

4. Simplify: cancel out all denominators of mini-fractions with associated common factors.

*The rational expression is now simple.

4a. Multiply, distribute, combine like-terms in numerator.

4b. Multiply, distribute, combine like-terms in denominator.

4c. Put all terms in descending order.

5. Completely factor the numerator and denominator, separately.

6. Cancel out any common factors from numerator & denominator.

Simplify Overall Numerator & Overall Denominator Separately Method (short version)

0. Put whole factors over 1 and put all overall numerators & denominators in parentheses.

1. Factor all numerators & denominators.

2a. Find the LCD of the fractions in the overall-numerator.

2b. Find the LCD of the fractions in the overall-denominator.

3a. Convert and add fractions in overall-numerator.

3b. Convert and add fractions in overall-denominator.

4a. Leave all denominators of mini-fractions as un-multiplied factors.

4b. Simplify all numerators of mini-fractions.

4c. Factor both numerators of the two remaining mini-fractions.

5. Divide the top fraction by the bottom fraction.

6. Simplify the one remaining fraction.

Note 4: Using the LCD is similar, yet distinctly different in the All-LCD-Method than for addition/subtraction of rational expressions in the overall-numerator & overall-denominator method. In addition/subtraction of rational expressions, the missing factors of the LCD are multiplied times the numerator and denominator of each fraction to convert the fractions into like-fractions, whereas

for simplification of complex rational expressions, the whole LCD is multiplied times the numerator only of each mini-fraction.

Annotated Example 1 Using the All-LCD Method

Write out the problem leaving plenty of room on both sides and below:

1. Put 4 over 1, and put all polynomials in parentheses:

1.  The numerators are already simplified and cannot be factored. All denominators except “x2 – 4” cannot be factored. Notice that “x2 – 4” is the difference of two squares. Factor it into conjugate pair binomials:

2. Find the LCD of all mini fractions and write it to the side.

LCD = x(x – 2)(x + 2)

3. Multiply all mini fractions by the LCD:

Cross out the common factors:

  And remove the crossed-out common factors. Notice that all denominators of the mini-fractions will be eliminated (also, remove the denominator “1”) and the expression will change from complex to simple. It will look like:

4. Simplify both the numerator and denominator, separately, by multiplying, distributing, then combining like-terms

4a. These steps show the multiplication:

4b. This step shows combining like-terms and arranging into descending order:

5. Completely factor the numerator and denominator, separately. In the numerator, the GCF is -3x. The denominator is a trinomial, so try to factor it into two binomials.

The denominator cannot be further factored.

Note: the numerator could have automatically been factored to this from step 4a either by using “(x2 – 4)” as a GCF, or by combining like-terms, but this will not always be an option.

6. Look for any common factors in the numerator and denominator. In this case, there are none, so the last step is the most simplified form.

Annotated Example 2 Using the Overall Numerator & Denominator Method

Write out the problem and leave plenty of room on the sides of each term:

1. Put 4 over 1. In this case, there are no polynomials to put parenthesis around in any mini-fraction:

1.  Since there are no polynomials in the numerators, they can’t be factored. The denominators in the top fractions can’t be factored, however the denominators of the bottom fractions can be factored (into exponential form in anticipation of finding the LCD):

2. Find the LCDs:

1. Find the LCD of the top fractions. It is “x”.

2. Find the LCD of the bottom fractions. It is “8”.

3. Convert fractions into like-fractions, then add:

1. In the overall-numerator, then

2. In the overall-denominator.

4.  

1. Leave denominator factors un-multiplied.


2. Simplify all numerators of mini factors. In this case, they are already simplified.

3. Factor the numerators of both mini-fractions. The numerator of the bottom fraction can’t be factored.

5. Divide the top fraction by the bottom fraction by inverting and multiplying:

There are no common factors to cancel out, so the simplified form is:

The Wrong Way to Simplify a Rational Expression

This section highlights a serious mistake that students make all the time. It involves the last step of simplifying a rational expression. Oddly enough, students often perform the more difficult part of the problem correctly before getting to this step, which is why I believe students commit this mistake more out of laziness than ignorance. Regardless of why, it must be prevented, especially because this is often the last step in a problem (and if you have an instructor that doesn’t give partial credit, this step could make or break a problem). Here are examples of the wrong and right way to simplify a rational expression. The step(s) I’m highlighting in this section are the same seen insteps 5 & 6 of The All-LCD Method for Simplifying Rational Expressions.

What you need to realize is: you can’t factor out a term in the numerator with a term in the denominator when (and because) terms are separated by “+” and “-“ signs. You can only cancel factors in the numerator with factors in the denominator… and factors are multiplied, not added or subtracted, together.

The wrong thing to do is to instantly cancel out a factor in the numerator with a term or factor in the denominator, without first factoring the numerator (either factoring the GCF out or factoring it into smaller polynomials), and taking into consideration the significance of the plus or minus sign on top between the top terms.

Let’s start with this example, which contains a binomial in the numerator and a monomial in the denominator:

The following is the wrong way:

in which one attempts to factor 3x out of the 12x2 (to 4x) and the 3x (to 1).

Alternatively, the following is also the wrong way:

in which one attempts to factor 3 out of -6 (to -2) and out of 3x (to 1, leaving x).

Notice how, in each wrong way example, the terms incorrectly cancelled out have plus or minus signs in front or behind them. This is the key sign (no pun intended) that should tell you not to cancel out terms.

The Correct Way:

You need to look for a GCF in the numerator, which in this case is 6, and then factor it out. At this point, the common factor of 3 can be cancelled out of the numerator (6) and denominator (3x), as shown:

The expression above on the right can be considered the most simplified form. Compare this answer to the wrong answers from before. As I mentioned in the notes at the end of the All-LCD Method for Simplifying Rational Expressions, the final answer can be shown like this, or by distributing (multiplying) the factor of 2 through the (2x2 – 1) in the numerator. Since simplification often involves complete factorization and not the reverse (multiplying), I believe this form is the most simplified. If you choose to multiply through (perhaps at the suggestion of your instructors – you should always report the answer the way they prefer it, since they’re grading you), it will appear like this:

or, if you break it apart into separate fractions:

which will then simplify to:

Again, find out from your instructors how they want you to report your answer.

Let’s look at another example, one with a trinomial in the numerator and a binomial in the denominator:

The Wrong Way:

There are many wrong ways to approach a such problem. One wrong way might be to attempt to cancel out x from 3x2 (to 3x) in the numerator and from 2x (to 2) in the denominator. Another wrong thing to do would be to factor 3 out of -15x (to –5x) in the numerator and out of -6 (to -2) in the denominator. If those erroneous cancellations were performed, it would wrongly give:

… which would conclude to be undefined.

The Correct Way:

Going back to the original example, factor the GCF (which is 3) out of the numerator. Then, factor the GCF (which is 2) out of the denominator, which would make:

Next, go back to the numerator and see if the trinomial inside parentheses can be factored, which it can be, into the two binomials, seen below. It is revealed that the common factor in the numerator and denominator to be cancelled out is (x – 3), shown below:

After (x – 3) is cancelled out, the final simplified form is:

Compare this to the wrong answer shown above.

Extraneous Solutions

It is important you know what extraneous solutions are, when to look for them, and how to deal with them, because they are tricky and deceptive things. Extraneous solutions (also commonly known as extraneous roots) appear to be solutions to a problem you just solved, but actually aren’t. They tend to come from the following two places:

·   A (variable in the) denominator, and

·   A (variable inside a) radical.

Based on the location of variables in an equation, these can be thought of as solution-exceptions, for the following reasons:

·   Any fraction whose denominator is zero is undefined.

·   Also, anytime a radicand (of any even root) is negative, the result is not real (still counts as undefined).

You can find extraneous solutions in one of two ways.

·   One is by checking all answers after you’ve solved for the unknowns.

·   Another is by finding it (or them) first, before solving the equation.

o I recommend this way, as explained in the next section; it’s easier, and this way, if you forget to do the check step at the end of a problem, as many people do, it won’t matter. The method for finding extraneous roots in radicals is shown in the Radicals, Roots & Powers section.


Procedure for Solving Equations with Rational Expressions & Extraneous Solutions

A. Find extraneous solutions (solution-exceptions) first [find all possible values of x that would make the denominator (any denominator in the problem) = 0]. When you get to the end of the problem, compare your solutions to the exceptions, and eliminate the extraneous solutions from your answers. To do this:

A0. Write out the denominators only (separately, if more than one fraction).

A1. Factor each denominator.

A2. Set all denominator factors = 0 and solve for x (or whatever the variable is).

A3. Put a slash though the = sign, to remind yourself that x does not equal the number(s) just determined. Save these off to the side to refer to them at the end of Part B.

B. Solving the Problem:

B0. Write out the whole problem. Write the denominators as the factors you determined through factoring from step A1.

B1. Determine the LCD.

B2. Multiply the LCD times each (numerator only of each) fraction and non-fraction-term (on both sides of the equation). This will eliminate all denominators (and thus all fractions).

B3. Simplify (combine like-terms) and solve using the Procedure for Solving a Simple Algebraic Equation with One Variable).

B4. Compare your answers to those found in step A3 and cross out any extraneous solutions.

Cross Multiplication  

Cross Multiplication is the act of multiplying the numerator of one fraction times the denominator of the fraction on the other side of the equal sign, and vice versa.

Cross multiplication is commonly used when doing problems involving proportions, specifically when there is one fraction (only) on each side of the equal sign.

When should you use it? You should use it when trying to solve for a variable in the denominator and when there is only one fraction on each side of the equal sign.

You can only cross multiply if there is only one fraction on each side of the equal sign or you simply can’t do it. However…

·   If you have more than one fraction on either side of the equal sign, you can either:

o Move one of the fractions to the other side (you can do this if you have two fractions on the same side equal to zero on the other side), or:

o Find the LCD of all fractions & multiply all fractions by the LCD. This will then eliminate all denominators and you will no longer have to do cross-multiplication.

Don’t be fooled. For cross multiplication to occur, there must be one fraction on each side of the equal sign, however, the numerators & denominators themselves can be polynomials (if they are, multiply accordingly). Also, you can easily convert a whole number or polynomial into a fraction by putting it over “1”.

A mistake students commonly make is trying to cross multiply fractions that are on the same side of the equal sign. Cross multiplication can only be performed across equal signs.

See in the example below how:

is cross-multiplied to become

(2)(x) = (5)(3)

which becomes 2x = 15

and can be solved by dividing both sides by the coefficient 2:

, and thus or 7.5

Cross-multiplication is not the same as multiplying fractions (on the same side of the equal sign). When fractions are on the same side, multiply the numerators by numerators and the denominators by denominators (see: Multiplying Fractions). Also,

Cross-Multiplication is different than “Cross-Cancelling.”

Cross-Multiplication vs. Cross Cancelling

It is important to use these methods at the appropriate times and to use the terminology correctly, as they are completely different.

Cross-multiplication is done when you have one fraction set equal another fraction, one of which contains an unknown variable. This is often seen when doing work with proportions and sometimes percent problems. Cross multiplication is and can only be performed across an “=” sign by multiplying the numerator of the left fraction by the denominator of the right fraction, and setting that product equal to the product of multiplying the denominator on the left times the numerator on the right. This is explained in more detail in the previous section.

Cross canceling is a simplification technique. It is the process of simplifying and reducing fractions by canceling out common factors in the numerator of one fraction with the denominator of itself or another fraction it is multiplied by.

See in the example below how

when factored, which reduces to

because

·   one of the top 2s cross-cancels with the 2 in the bottom of the other fraction, and

·   the 5 in the top cross-cancels with the 5 in the bottom of the other fraction,

which then equals after you multiply the fractions.