Arithmetic Reasoning and Mathematics Knowledge Review: Algebra

Terms and Coefficients
Mathematical expressions consist of a combination of one or more values arranged in terms that are added together. As such, an expression could be just a single number, including zero. A variable term is the product of a real number, also called a coefficient, and one or more variables, each of which may be raised to an exponent. Expressions may also include numbers without a variable, called constants or constant terms. The expression

, for example, is a single term where the coefficient is the real number 6 and the variable is

. Note that if a term is written as simply a variable to some exponent, like

, then the coefficient is 1, because

.

Linear Expressions
A single variable linear expression is the sum of a single variable term, where the variable has no exponent, and a constant, which may be zero. For instance, the expression

 has
2w as the variable term and 7 as the constant term. It is important to realize that terms are separated by addition or subtraction. Since an expression is a sum of terms, expressions such as

 can be written as

 to emphasize that the constant term is negative. A real-world example of a single variable linear expression is the perimeter of a square, four times the side length, often expressed:

.
In general, a linear expression is the sum of any number of variable terms so long as none of the variables have an exponent. For example,

 is a linear expression, but

 is not. In the same way, the expression for the perimeter of a general triangle, the sum of the side lengths:

, is considered to be linear, but the expression for the area of square, the side length squared:

, is not.

Linear Equations
Equations that can be written as

, if we solve for

 we get a solution of

. In other words, the root of the equation is -2. This is found by first subtracting 10 from both sides, which gives

. Next, simply divide both sides by the coefficient of the variable, in this case 5, to get

. This can be checked by plugging -2 back into the original equation

.
The solution set is the set of all solutions of an equation. In our example, the solution set would simply be -2. If there were more solutions (there usually are in multivariable equations) then they would also be included in the solution set. When an equation has no true solutions, this is referred to as an empty set. Equations with identical solution sets are equivalent equations. An identity is a term whose value or determinant is equal to 1.
Linear equations can be written many ways. Below is a list of some forms linear equations can take:

Standard Form:

; the slope is

 and the y-intercept is



· Slope
Intercept Form:

, where m is the slope and

 is a point on the line
Two-Point Form:

 , where

 and

 are two points on the given line
Intercept Form:

, where

 is the point at which a line intersects the x-axis, and

 is the point at which the same line intersects the y-axis

Solving One-Variable Linear Equations
Multiply all terms by the lowest common denominator to eliminate any fractions.  Look for addition or subtraction to undo so you can isolate the variable on one side of the equal sign.  Divide both sides by the coefficient of the variable.  When you have a value for the variable, substitute this value into the original equation to make sure you have a true equation. Consider the following example:
Kim's savings are represented by the table below. Represent her savings, using an equation.

X (Months) - Y (Total Savings)
2- $1300
5 - $2050
9 - $3050
11 - $3550
16 - $4800

The table shows a function with a constant rate of change, or slope, of 250. Given the points on the table, the slopes can be calculated as

,

,

, and

, each of which equals 250. Thus, the table shows a constant rate of change, indicating a linear function. The slope-intercept form of a linear equation is written as

, where m represents the slope and b represents the y-intercept. Substituting the slope into this form gives

. Substituting corresponding x- and y-values from any point into this equation will give the y-intercept, or b. Using the point, (2, 1300), gives

 , which simplifies as b = 800. Thus, her savings may be represented by the equation,

.

Rules for Manipulating Equations

Like Terms
Like terms are terms in an equation that have the same variable, regardless of whether or not they also have the same coefficient. This includes terms that lack a variable; all constants (i.e. numbers without variables) are considered like terms. If the equation involves terms with a variable raised to different powers, the like terms are those that have the variable raised to the same power.
For example, consider the equation

. In this equation, 2 and –7 are like terms; they are both constants.

,

, and 2

 are like terms: they all include the variable

 raised to the first power.

 and

 are like terms; they both include the variable

, raised to the second power.

 and

 are not like terms; although they both involve the variable

, the variable is not raised to the same power in both terms. The fact that they have the same coefficient, 2, is not relevant.


Carrying Out the Same Operation on Both Sides of an Equation
When solving an equation, the general procedure is to carry out a series of operations on both sides of an equation, choosing operations that will tend to simplify the equation when doing so. The reason why the same operation must be carried out on both sides of the equation is because that leaves the meaning of the equation unchanged, and yields a result that is equivalent to the original equation. This would not be the case if we carried out an operation on one side of an equation and not the other. Consider what an equation means: it is a statement that two values or expressions are equal. If we carry out the same operation on both sides of the equation—add 3 to both sides, for example—then the two sides of the equation are changed in the same way, and so remain equal. If we do that to only one side of the equation—add 3 to one side but not the other—then that wouldn't be true; if we change one side of the equation but not the other then the two sides are no longer equal.

Advantage of Combining Like Terms
Combining like terms refers to adding or subtracting like terms—terms with the same variable—and therefore reducing sets of like terms to a single term. The main advantage of doing this is that it simplifies the equation. Often combining like terms can be done as the first step in solving an equation, though it can also be done later, such as after distributing terms in a product.

For example, consider the equation

. The 2 and the 3 in the second set of parentheses are like terms, and we can combine them, yielding

. Now we can carry out the multiplications implied by the parentheses, distributing outer 2 and 3 accordingly:

. The

 and the

 are like terms, and we can add them together:

. Now, the constants
6, 15, and –4 are also like terms, and we can combine them as well: subtracting
6 and 15 from both sides of the equation, we get

, or

, which simplifies further to

.

Canceling Terms on Opposite Sides of an Equation
Two terms on opposite sides of an equation can be canceled if and only if they exactly match each other. They must have the same variable raised to the same power and the same coefficient. For example, in the equation

,

 appears on both sides of the equation, and can be canceled, leaving

. The 6 on each side of the equation cannot be canceled, because it is added on one side of the equation and subtracted on the other. While they cannot be canceled, however, the 6 and –6 are like terms and can be combined, yielding

, which simplifies further to

.
It's also important to note that the terms to be canceled must be independent terms and cannot be part of a larger term. For example, consider the equation

. We cannot cancel the

s, because even though they match each other they are part of the larger terms

 and

. We must first distribute the 2 and 3, yielding

. Now we see that the terms with the

's do not match, but the 12's do, and can be canceled, leaving

, which simplifies to

.

Process for Manipulating Equations

Isolating Variables
To isolate a variable means to manipulate the equation so that the variable appears by itself on one side of the equation, and does not appear at all on the other side. Generally, an equation or inequality is considered to be solved once the variable is isolated and the other side of the equation or inequality is simplified as much as possible. In the case of a two-variable equation or inequality, only one variable need be isolated; it will not usually be possible to simultaneously isolate both variables.
For a linear equation—an equation in which the variable only appears raised to the first power—isolating a variable can be done by first moving all the terms with the variable to one side of the equation and all other terms to the other side. (Moving a term really means adding the inverse of the term to both sides; when a term is moved to the other side of the equation its sign is flipped.) Then combine like terms on each side. Finally, divide both sides by the coefficient of the variable, if applicable. The steps need not necessarily be done in this order, but this order will always work.

Equations with More Than One Solution
Some types of non-linear equation, such as equations involving squares of variables, may have more than one solution. For example, the equation

 has two solutions: 2 and –2. Equations with absolute values can also have multiple solutions:

 has the solutions

 and

.
It is also possible for a linear equation to have more than one solution, but only if the equation is true regardless of the value of the variable. In this case, the equation is considered to have infinitely many solutions, because any possible value of the variable is a solution. We know a linear equation has infinitely many solutions if when we combine like terms the variables cancel, leaving a true statement. For example, consider the equation

 Distributing, we get

; combining like terms gives

, and the

 terms cancel to leave

. This is clearly true, so the original equation is true for any value of

. We could also have canceled the 10s leaving

, but again this is clearly true—in general if both sides of the equation match exactly, it has infinitely many solutions.

Equations with No Solution
Some types of non-linear equations, such as equations involving squares of variables, may have no solution. For example, the equation

 has no solutions in the real numbers, because the square of any real number must be positive.
Similarly,

 has no solution, because the absolute value of a number is always positive.
It is also possible for an equation to have no solution even if does not involve any powers greater than one or absolute values or other special functions. For example, the equation

 has no solution. We can see that if we try to solve it: first we distribute, leaving

. But now if we try to combine all the terms with the variable, we find that they cancel: we have

 on the left and

 on the right, canceling to leave us with

. This is clearly false. In general, whenever the variable terms in an equation cancel leaving different constants on both sides, it means that the equation has no solution. (If we are left with the same constant on both sides, the equation has infinitely many solutions instead.)

Features of Equations That Require Special Treatment

Linear Equations
A linear equation is an equation in which variables only appear by themselves: not multiplied together, not with exponents other than one, and not inside absolute value signs or any other functions. For example, the equation

.

 is not a linear equation, because it involves a square root.

 is not a linear equation because even though there's no exponent on the

 directly, it appears as part of an expression that is squared. The two-variable equation

 is not a linear equation because it includes the term

, where two variables are multiplied together.
Linear equations can always be solved (or shown to have no solution) by combining like terms and performing simple operations on both sides of the equation. Some non-linear equations can also be solved by similar methods, but others may require more advanced methods of solution, if they can be solved analytically at all.

Solving Equations Involving Roots
In an equation involving roots, the first step is to isolate the term with the root, if possible, and then raise both sides of the equation to the appropriate power to eliminate it. Consider an example equation,

. In this case, begin by adding 1 to both sides, yielding

, and then dividing both sides by 2, yielding

. Now square both sides, yielding

. Finally, subtracting 1 from both sides yields

.
Squaring both sides of an equation may, however, yield a spurious solution—a solution to the squared equation that is not a solution of the original equation. It's therefore necessary to plug the solution back into the original equation to make sure it works. In this case, it does:

.
The same procedure applies for roots other than square roots. For example, given the equation

, we can first subtract 3 from both sides, yielding

 and isolating the root. Raising both sides to the third power yields

, i.e.

. We can now divide both sides by 2 to get

.

Solving Equations with Exponents
To solve an equation involving an exponent, the first step is to isolate the variable with the exponent. We can then take the appropriate root of both sides to eliminate the exponent. For instance, for the equation

, we can subtract

 from both sides to get

, and then subtract
17 from both sides to get

. Finally, we can divide both sides by –3 to get

. Finally, we can take the cube root of both sides to get

.
One important but often overlooked point is that equations with an exponent greater than 1 may have more than one answer. The solution to

 isn't simply

; it's

: that is,

 or

. For a slightly more complicated example, consider the equation

. Adding one to both sides yields

; taking the square root of both sides yields

. We can then add 1 to both sides to get

. However, there's a second solution: we also have the possibility that

, in which case

. Both

 and

 are valid solutions, as can be verified by substituting them both into the original equation.

Solving Equations with Absolute Values
When solving an equation with an absolute value, the first step is to isolate the absolute value term. We then consider the two possibilities: when the expression inside the absolute value is positive or when it is negative. In the former case, the expression in the absolute value equals the expression on the other side of the equation; in the latter, it equals the additive inverse of that expression—the expression times negative one. We consider each case separately, and finally check for spurious solutions.
For instance, consider solving

 for

. We can first isolate the absolute value by moving the

 to the other side:

. Now, we have two possibilities. First, that

 is positive, and hence

. Rearranging and combining like terms yields

, and hence

. The other possibility is that

 is negative, and hence

. In this case, rearranging and combining like terms yields

. Substituting

 and

 back into the original equation, we see that they are both valid solutions.
Note that the absolute value of a sum or difference applies to the sum or difference as a whole, not to the individual terms: in general,

 is not equal to

 or to

.


Spurious Solutions
A spurious solution may arise when we square both sides of an equation as a step in solving it, or under certain other operations on the equation. It is a solution to the squared or otherwise modified equation that is not a solution of the original equation. To identify a spurious solution, it's useful when you solve an equation involving roots or absolute values to plug the solution back into the original equation to make sure it's valid.

Choosing Which Variable to Isolate in Two-Variable Equations
Similar to methods for a one-variable equation, solving a two-variable equation involves isolating a variable: manipulating the equation so that a variable appears by itself on one side of the equation, and not at all on the other side. However, in a two-variable equation, you will usually only be able to isolate one of the variables; the other variable may appear on the other side along with constant terms, or with exponents or other functions.
Often one variable will be much more easily isolated than the other, and therefore that's the variable you should choose. If one variable appears with various exponents, and the other only raised it to the first power, the latter variable is the one to isolate: given the equation
y, the convention is that y is the independent variable.

P1. Seeing the equation

, a student divides the first terms on each side by 2, yielding

, and then combines like terms to get

. However, this is incorrect, as can be seen by substituting –3 into the original equation.
Explain what is wrong with the student's reasoning.
P2. Describe the steps necessary to solve the equation

.
P3. Describe the steps necessary to solve the equation

.
P4. Find all real solutions to the equation

.
P5. Find all real solutions to the equation

.
P6. Solve for

:

.
P7. Ray earns $10 an hour at his job.  Write an equation for his earnings as a function of time spent working.  Determine how long Ray has to work in order to earn $360.
P8. Simplify the following:


P1. As stated, it's easy to verify that the student's solution is incorrect:

 and

; clearly

.
The mistake was in the first step, which illustrates a common type of error in solving equations. The student tried to simplify the two variable terms by dividing them by 2. However, it's not valid to multiply or divide only one term on each side of an equation by a number; when multiplying or dividing, the operation must be applied to every term in the equation. So, dividing by 2 would yield not

.
P2. Our ultimate goal is to isolate the variable,

.
To that end we first move all the terms containing

 to the left side of the equation, and all the constant terms to the right side.
Note that when we move a term to the other side of the equation its sign changes. We are therefore now left with

.
Next, we combine the like terms on each side of the equation, adding and subtracting the terms as appropriate. This leaves us with

. A. this point, we're almost done; all that remains is to divide both sides by

 to leave the

 by itself. We now have our solution,

. We can verify that this is a correct solution by substituting it back into the original equation.
P3. Generally, in equations that have a sum or difference of terms multiplied by another value or expression, the first step is to multiply those terms, distributing as necessary:

, and

. So, the equation becomes

. We can now add

 to both sides to eliminate the variable from the right-hand side:

. Similarly, we can subtract 10 from both sides to move all the constants to the right:

.
Finally, we can divide both sides by 9, yielding the final answer,

.
P4. It's not hard to isolate the root: subtract one from both sides, yielding

. Finally, multiply both sides by –1, yielding

. Squaring both sides of the equation yields

. However, if we plug this back into the original equation, we get

, which is false.
Therefore

 is a spurious solution, and the equation has no real solutions.
P5. This equation has two possibilities:

, which simplifies to

; or

, which simplifies to

. However, if we try substituting both values back into the original equation, we see that only

 yields a true statement.

 is a spurious solution;

 is the only valid solution to the equation.
P6. Start by isolating the term with the root. We can do that by moving the

 and the 1 to the other side, yielding

, or

. Dividing both sides of the equation by 2 would give us a fractional term that could be messy to deal with, so we won't do that for now. Instead, we square both sides of the equation; note that on the left-hand side the 2 is outside the square root sign, so we have to square it. As a result, we get

. Expanding both sides gives us

. In this case, we see that we have

 on both sides, so we can cancel the

 (which is what allows us to solve this equation despite the different powers of

).
We now have

, or

. Since the variable is raised to an even power, we need to take the positive and negative roots, so

: that is,

 or

. Substituting both values into the original equation, we see that

 satisfies the equation but

 does not; hence

 is a spurious solution, and the only solution to the equation is

.
P7. The number of dollars that Ray earns is dependent on the number of hours he works, so earnings will be represented by the dependent variable y and hours worked will be represented by the independent variable x.  He earns 10 dollars per hour worked, so his earnings can be calculated as

.
To calculate the number of hours Ray must work in order to earn $360, plug in
360 for y and solve for x:





P8. To simplify this equation, we must isolate one of its variables on one side of the equation. In this case, the

 appears under an absolute value sign, which makes it difficult to isolate. The y, on the other hand, only appears without an exponent—the equation is linear in y.
We will therefore choose to isolate the y. The first step, then, is to move all the terms with y to the left side of the equation, which we can do by subtracting 5y from both sides:


We can then move all the terms that do not include y to the right side of the equation, by subtracting

 and
2 from both sides of the equation:


Finally, we can isolate the y by dividing both sides by –3.


This is as far as we can simplify the equation; we cannot combine the terms inside and outside the absolute value sign. We can therefore consider the equation to be solved.

Inequalities
Commonly in algebra and other upper-level fields of math you find yourself working with mathematical expressions that do not equal each other. The statement comparing such expressions with symbols such as < (less than) or > (greater than) is called an inequality. An example of an inequality is

.
To solve for

, simply divide both sides by

 and the solution is shown to be

. Graphs of the solution set of inequalities are represented on a number line. Open circles are used to show that an expression approaches a number but is never quite equal to that number.

Conditional inequalities are those with certain values for the variable that will make the condition true and other values for the variable where the condition will be false. Absolute inequalities can have any real number as the value for the variable to make the condition true, while there is no real number value for the variable that will make the condition false. Solving inequalities is done by following the same rules as for solving equations with the exception that when multiplying or dividing by a negative number the direction of the inequality sign must be flipped or reversed. Double inequalities are situations where two inequality statements apply to the same variable expression. An example of this is

.

Determining Solutions to Inequalities
To determine whether a coordinate is a solution of an inequality, you can substitute the values of the coordinate into the inequality, simplify, and check whether the resulting statement holds true. For instance, to determine whether

 is a solution of the inequality

, substitute the values into the inequality,

. Simplify the right side of the inequality and the result is

, which is a false statement. Therefore, the coordinate is not a solution of the inequality. You can also use this method to determine which part of the graph of an inequality is shaded. The graph of

 includes the solid line

and, since it excludes the point

 to the left of the line, it is shaded to the right of the line.

Flipping Inequality Signs
When given an inequality, we can always turn the entire inequality around, swapping the two sides of the inequality and changing the inequality sign. For instance,

 is equivalent to

.
Aside from that, normally the inequality does not change if we carry out the same operation on both sides of the inequality. There is, however, one principal exception: if we multiply or divide both sides of the inequality by a negative number, the inequality is flipped. For example, if we take the inequality

 or

' is a compound inequality, as is '

 and

.' A. and inequality can be written more compactly by having one inequality on each side of the common part: '

 and

,' can also be written as

.
In order for the compound inequality to be meaningful, the two parts of an and inequality must overlap; otherwise no numbers satisfy the inequality. On the other hand, if the two parts of an or inequality overlap, then all numbers satisfy the inequality and as such is usually not meaningful.
Solving a compound inequality requires solving each part separately. For example, given the compound inequality '

 or

,' the first inequality,

, reduces to

, and the second part,

, reduces to

, so the whole compound inequality can be written as '

 or

.' Similarly,

 can be solved by dividing each term by 2, yielding

.

Solving Inequalities Involving Absolute Values
To solve an inequality involving an absolute value, first isolate the term with the absolute value. Then proceed to treat the two cases separately as with an absolute value equation, but flipping the inequality in the case where the expression in the absolute value is negative (since that essentially involves multiplying both sides by

.)
The two cases are then combined into a compound inequality; if the absolute value is on the greater side of the inequality, then it is an or compound inequality, if on the lesser side, then it's an and.
Consider the inequality

. We can isolate the absolute value term by subtracting 2 from both sides:

. Now, we're left with the two cases

 or

: note that in the latter, negative case, the inequality is flipped.

 reduces to

, and

 reduces to

. Since in the inequality

 the absolute value is on the greater side, the two cases combine into an or compound inequality, so the final, solved inequality is '

 or

.'


Solving Inequalities Involving Square Roots
Solving an inequality with a square root involves two parts.
First, we solve the inequality as if it were an equation, isolating the square root and then squaring both sides of the equation. Second, we restrict the solution to the set of values of

 for which the value inside the square root sign is non-negative.
For example, in the inequality,

, we can isolate the square root by subtracting 1 from both sides, yielding

. Squaring both sides of the inequality yields

, so

.
Since we can't take the square root of a negative number, we also require the part inside the square root to be non-negative. In this case, that means

.
Adding 2 to both sides of the inequality yields

.
Our final answer is a compound inequality combining the two simple inequalities:

 and

, or

.
Note that we only get a compound inequality if the two simple inequalities are in opposite directions; otherwise we take the one that is more restrictive.
The same technique can be used for other even roots, such as fourth roots. It is not, however, used for cube roots or other odd roots—negative numbers do have cube roots, so the condition that the quantity inside the root sign cannot be negative does not apply.

Special Circumstances
Sometimes an inequality involving an absolute value or an even exponent is true for all values of

, and we don't need to do any further work to solve it. This is true if the inequality, once the absolute value or exponent term is isolated, says that term is greater than a negative number (or greater than or equal to zero). Since an absolute value or a number raised to an even exponent is always non-negative, this inequality is always true.
P1. Analyze the following inequalities:
(a)



(b)


P1. (a) Subtracting 2 from both sides yields

; multiplying by

—and flipping the inequality, since we're multiplying by a negative number—yields

. But since the absolute value cannot be negative, it's always greater than –1, so this inequality is true for all values of

.
(b) Subtracting 7 from both sides yields

; dividing by 2 yields

. But

 must be nonnegative, and hence cannot be less than or equal to –3; this inequality has no solution.

Polynomials
Polynomials are made up of monomials and polynomials. A monomial is a single variable or product of constants and variables, such as x, 2x, or

 . There will never be addition or subtraction symbols in a monomial. Like monomials have like variables, but they may have different coefficients. Polynomials are algebraic expressions which use addition and subtraction to combine two or more monomials. Two terms make a binomial, three terms make a trinomial, etc. The degree of a monomial is the sum of the exponents of the variables. The degree of a polynomial is the highest degree of any individual term.

Simplifying Polynomials
Simplifying polynomials requires combining like terms. The like terms in a polynomial expression are those that have the same variable raised to the same power. It is often helpful to connect the like terms with arrows or lines in order to separate them from the other monomials. Once you have determined the like terms, you can rearrange the polynomial by placing them together. Remember to include the sign that is in front of each term. Once the like terms are placed together, you can apply each operation and simplify. When adding and subtracting polynomials, only add and subtract the coefficient, or the number part; the variable and exponent stay the same.

The FOIL Method
In general, multiplying polynomials is done by multiplying each term in one polynomial by each term in the other and adding the results. In the specific case for multiplying binomials, there is useful acronym, FOIL, that can help you make sure to cover each combination of terms. The FOIL method for

 would be:

F - Multiply the first terms of each binomial -


O - Multiply the outer terms -


I - Multiply the inner terms -


L - Multiply the last terms of each binomial -


Then add up the result of each and combine like terms:

.
For example, using the FOIL method on binomials

 and







This results in:


Combine like terms:




Dividing Polynomials
To divide polynomials, set up a long division problem, dividing a polynomial by either a monomial or another polynomial of equal or lesser degree.

When dividing by a monomial, divide each term of the polynomial by the monomial.

When dividing by a polynomial, begin by arranging the terms of each polynomial in order of one variable. You may arrange in ascending or descending order, but be consistent with both polynomials. To get the first term of the quotient, divide the first term of the dividend by the first term of the divisor. Multiply the first term of the quotient by the entire divisor and subtract that product from the dividend. Repeat for the second and successive terms until you either get a remainder of zero or a remainder whose degree is less than the degree of the divisor. If the quotient has a remainder, write the answer as a mixed expression in the form:



For example, we can evaluate the following expression in the same way as long division:













































45



When factoring a polynomial, first check for a common monomial factor, that is look to see if each coefficient has a common factor or if each term has an

 in it. If the factor is a trinomial but not a perfect trinomial square, look for a factorable form, such as one of these:






For factors with four terms, look for groups to factor. Once you have found the factors, write the original polynomial as the product of all the factors. Make sure all of the polynomial factors are prime. Monomial factors may be prime or composite. Check your work by multiplying the factors to make sure you get the original polynomial.

Below are patterns of some special products to remember to help make factoring easier:

Perfect trinomial squares:

 or


· Difference between two squares:


·     
Sum of two cubes:


- Note: the second factor is not the same as a perfect trinomial square, so do not try to factor it further.
· Difference between two cubes:



- Again, the second factor is not the same as a perfect trinomial square.
·        
Perfect cubes:

 and



Rational expressions
Rational expressions are fractions with polynomials in both the numerator and the denominator; the value of the polynomial in the denominator cannot be equal to zero. Be sure to keep track of values that make the denominator of the original expression zero as the final result inherits the same restrictions. For example, a denominator of

 indicates that the expression is not defined when

 and as such, regardless of any operations done to the expression, it remains undefined there.

To add or subtract rational expressions, first find the common denominator, then rewrite each fraction as an equivalent fraction with the common denominator. Finally, add or subtract the numerators to get the numerator of the answer, and keep the common denominator as the denominator of the answer.

When multiplying rational expressions factor each polynomial and cancel like factors (a factor which appears in both the numerator and the denominator). Then, multiply all remaining factors in the numerator to get the numerator of the product, and multiply the remaining factors in the denominator to get the denominator of the product. Remember: cancel entire factors, not individual terms.

To divide rational expressions, take the reciprocal of the divisor (the rational expression you are dividing by) and multiply by the dividend.

Simplifying Rational Expressions
To simplify a rational expression, factor the numerator and denominator completely. Factors that are the same and appear in the numerator and denominator have a ratio of 1. For example, look at the following expression:


The denominator,

, is a difference of squares. It can be factored as

. The factor

 and the numerator

 are opposites and have a ratio of –1. Rewrite the numerator as

. So, the rational expression can be simplified as follows:


Note that since the original expression is only defined for

, the simplified expression has the same restrictions.

Rational Expressions

P1. Expand the following polynomials:
(a)


(b)


(c)


P2. Evaluate the following rational expressions:
(a)


(b)


P1. (a) Apply the FOIL method and the distributive property of multiplication:









(b) Note the difference of squares form:












(c) Multiply each pair of monomials and combine like terms:





P2. (a) Rather than trying to factor the fourth-degree polynomial, we can use long division:




























0


Note that since the original expression is only defined for

, the simplified
The denominator,

, is a difference of squares. It can be factored as

. The numerator,

, is a perfect square. It can be factored as

. So, the rational

expression is only defined for

, the simplified

Factorials
The factorial is a function that can be performed on any non-negative integer. It is represented by the ! sign written after the integer on which it is being performed. The factorial of an integer is the product of all positive integers less than or equal to the number. For example, 4! (read '4 factorial') is calculated as

.
Since 0 is not itself a positive integer, nor does it have any positive integers less than it, 0! cannot be calculated using this method. Instead, 0! is defined by convention to equal 1. This makes sense if you consider the pattern of descending factorials: