Expressions in standard form

There are many different types of algebraic expressions, which be classified by examing the operations used in them. Each type has a standard form; usually the first step in dealing with any expression is to put it in standard form.

Types of expressions

A constant expression is an algebraic expression with no variables in it; it's the same thing as an arithmetic expression. A literal expression is an algebraic expression with no constants or operations in it; it simply consists of a single variable. Most algebraic expressions, of course, are more complicated than these.

In this course, we'll mostly be dealing with linear expressions. A linear expression is an algebraic expression made up only from any or all of these:

• Constants;
• Variables;
• Multiplication by constants;
• Taking opposites (optional);
• Subtraction (optional);
• Division by nonzero constants (optional).
Note that the optional bits are really included in the previous items: taking the opposite is equivalent to multiplying by the constant −1, then subtraction is simply adding the opposite, and division by a nonzero constant is simply multiplication by some other constant (the reciprocal).

Typical linear expressions are things like x + 2, 2t + 4, and 5x − y. But notice some things that you can't do in a linear expression: you can't divide by a non-constant expression, or use exponentiation, or take absolute values. So for example, 1/x, y2, and |a| are not linear expressions.

A monomial is an algebraic expression made up only from any or all of these:

• Constants;
• Variables;
• Multiplication;
• Taking opposites (optional);
• Division by nonzero constants (optional);
• Raising to the powers of constant whole numbers (optional).
Again, the optional bits are really included in the previous items. For example, raising to the power of a constant whole number is either the constant 1 (if the exponent is 0), trivial (if the exponent is 1), or made up of multiplication (if the exponent is more than 1).

Typical monomials are things like x2, 2xy, and −xy3/5. But again some things you can't do in a monomial: you can't add or subtract, or divide by a non-constant expression, or raise to the power of a negative number, and you still can't take absolute values. So for example, x + 5, 1/x, y−2, and |a| are not monomials.

A polynomial is an algebraic expression made up only from any or all of these:

• Constants;
• Variables;
• Multiplication;
• Taking opposites (optional);
• Subtraction (optional);
• Division by nonzero constants (optional);
• Raising to the powers of constant whole numbers (optional).
Basically, anything you can do in a linear expression or a monomial is fair game in a polynomial. Again, the optional bits are really included in the previous items.

Typical polynomials include the linear expressions and monomials above, as well as things like x2 + 5 and 2xy2 + 3x. But there are still some things you can't do in a polynomial: you still can't divide by a non-constant expression, raise to the power of a negative number, or take absolute values. So for example, 1/x, y−2, and |a| are still not polynomials.

Finally, a rational expression is an algebraic expression made up only from any or all of these:

• Constants;
• Variables;
• Multiplication;
• Taking reciprocals;
• Taking opposites (optional);
• Subtraction (optional);
• Division (optional);
• Raising to the powers of constant integers (optional).
The name ‘rational expression’ comes from the fact that these are the same operations that you use in forming a rational number out of the constants 0 and 1. Again, the optional bits are really included in the previous items; for example, division is the same as multiplication by a reciprocal, and raising to the power of a negative integer is built out of the constant 1 and division.

Typical rational expressions include all the examples above, as well as things like 1/x, y−2, and even (2x2 + 5)/(4xy − 7z). But there are still a few things that you can't do in a rational expression; most of these are operations that you won't study until Intermediate Algebra, but you still can't take absolute values. So for example, |a| is still not a rational expression.

We really won't use rational expression in this course, except as occasional examples. We'll study polynomials for a while, but most of this course will be about linear expressions. However, we'll also study expressions that are almost linear but include absolute values; I don't know a special name for those.

Equivalent expressions

We often want to replace a complicated expression with a simpler one that means the same thing. For example, the expression x + 4 + 2 obviously means the same thing as x + 6, since 4 + 2 = 6. More interestingly, the expression x + x + 4 means the same thing as 2x + 4, because 2x is x + x when you think of multiplication as repeated addition. (Which of these is simpler depends on your point of view, but usually 2x + 4 is more convenient in Algebra.)

Two algebraic expressions are equivalent if they always lead to the same result when you evaluate them, no matter what values you substitute for the variables. For example, if x = 3, then x + x + 4 = 3 + 3 + 4 = 10 and 2x + 4 = 2(3) + 4 = 10 also. There's nothing special about 3 here; the same thing would happen no matter what value we used for x, so x + x + 4 is equivalent to 2x + 4. (That's really what I meant when I said above that they mean the same thing.) In other words, x + x + 2 = 2x + 2 for every value of x.

When I say that you get the same result, this includes the possibility that the result is undefined. For example, 1/x + 1/x is equivalent to 2/x; even when x = 0, they both come out the same (in this case, undefined). In other words, 1/x + 1/x = 2/x whenever either side is defined, which is the most that you could ask for. In contrast, x2/x is not equivalent to x; they usually come out the same, but they are different when x = 0. (Then x2/x is undefined, but x is 0.) So the best you can say in that case is that x2/x = x whenever the left-hand side is defined, or that x2/x = x for x ≠ 0.

Most of the identities that I mentioned before are equivalences of algebraic expressions. For example, a + b = b + a for any real numbers a and b. But to say that these are equal for any real numbers is simply to say that the expressions a + b and b + a are equivalent. On the other hand, the identity that a/a = 1 for any nonzero real number a is not quite an equivalence, since a cannot be zero.

So far, I've only used an identity to evaluate each expression for the same value of the variables, that is to substitute constants (specific numbers) for variables in the identity. So for example, if a = −3 and b = 4, then I get −3 + 4 = 4 + (−3); if you further remember that subtraction means adding the opposite, this tells you how to calculate −3 + 4 as 4 − 3, which is the number 1. But in fact, identities are good for more than that, and for Algebra we need to use them in more general ways. In fact, given any equivalence of algebraic expressions, you can get another equivalence by substituting (not necessarily a constant but) any defined algebraic expression for one of the variables (and you can do this multiple times to substitute for multiple variables). For example, if a = −3 (as before) but now b = 2x, then you get −3 + 2x = 2x − 3. This is also an example of the law that a + b = b + a, just like −3 + 4 = 4 − 3 is, but now it's an example that we'll need in Algebra.

Standard forms

For linear expressions, monomials, and polynomials, there is a notion of standard form that an expression may be in. (Actually, the most general notion —at least for this course— is the standard form of a polynomial; linear expressions and monomials are just special cases of this.)

A linear expression in standard form is (essentially) the sum of one or more terms: one constant term and one term for each variable in the expression. Furthermore, the term for a given variable must be the product of a constant coefficient and that variable, while the constant term (of course) must be a constant. Finally, the terms should come in alphabetical order (by the variable), with the constant term last. For example, in the linear expression 2x + 3y + 4, the terms are 2x, 3y, and 4. The first term, 2x, is the x-term, whose coefficient is 2. The next term, 3y, is the y-term, whose coefficient is 3. The constant term is 4; we consider 4 to be its own coefficient, on the grounds that 4 = 4 · 1.

There are some degenerate cases. If the coefficient on any of the variables is 1, then we can omit it; for example, 1x + 2 is equivalent to simply x + 2. If any coefficient is negative, then we use the opposite of a term with a positive coefficient, which means that (unless this is the first term) we write the sum using subtraction; for example, 5x + (−3)y + 4 is equivalent to simply 5x − 3y + 4. If the coefficient on any of the variables is −1, then we combine the previous two rules; for example, 5x + (−1)y + 4 is equivalent to simply 5x − y + 4. Finally, if any coefficient is 0, then we can omit the entire term; for example, 5x + 3y + 0 is equivalent to simply 5x + 3y. (The exception to this is that if every coefficient is 0, then we keep the constant term so that we have at least one term.) These simplifications are considered part of the standard form.

A monomial in standard form is (essentially) the product of one or more factors: a constant coefficient and one factor for each variable in the expression. Furthermore, the factor for a given variable must be the variable raised to the power of a constant whole number, the degree of that variable. Finally, the factors should come in alphabetical order (by the variable), with the coefficient (or constant factor) first. For example, in the monomial 4x2y3, the factors are 4, x2, and y3. First, the coefficient is 4. The next factor, x2, is the x-factor, whose degree is 2. The last factor, y3, is the y-factor, whose degree is 3. We consider the degree of the coefficient (in this case, 4) to be 0 on the grounds that 4 = 4x0 for any x.

Again, there are some degenerate cases. If the degree for any of the variables is 1, then we can omit it; for example, 4x1y3 is equivalent to simply 4xy3. Similarly, if the degree for any of the variables in 0, then we can omit the entire factor; for example, 4x0y3 is equivalent to simply 4y3. Alternatively, if the coefficient is 1, then we can omit it; for example, 1x2y3 is equivalent to simply x2y3. (The exception to this is that if every degree is 0 and the coefficient is 1, then we keep the coefficient so that we have at least one factor.) If the coefficient is negative, then we use the opposite of the monomial with a positive coefficient; for example, (−4)x2y3 is equivalent to simply −4x2y3. If the coefficient is −1, then we combine the previous two rules; for example, (−1)x2y3 is equivalent to simply −x2y3. Finally, if the coefficient is 0, then the entire monomial is zero; for example, 0x2y3 is equivalent to simply 0. Again, these simplifications are considered part of the standard form.

If the degree for a variable is 1, then we can leave the exponent off; for example, 3x1 is equivalent to simply 3x. If the degree for a variable is 0, then we can leave the variable out too; for example, 2x0 is equivalent to simply 2. If the coefficient is 1 (and there's at least one variable with a nonzero degree), then we can leave it out; for example, 1x3 is equivalent to simply x3. Finally, if the coefficient is −1 (and there's at least one variable with a nonzero degree), then we can replace it with a minus sign (indicating the additive inverse); for example, −1x3 is equivalent to simply −x3. These are also considered to be standard form; in fact, you should always perform these simplifications, at least when giving a final result.

The degree of a monomial is the sum of the degrees for the variables in the monomial. For example, the degree of 4x2y3 is 5, since 2 + 3 = 5. Also, the degree of 4xy3 is 4, since the degree for x is 1 and 1 + 3 = 4. Finally, the degree of a constant expression is 0, since there is nothing to add; remember that the degree of the coefficient itself is always zero.

Now, a polynomial in standard form is the sum of one or more terms; each term must be a monomial in standard form. Also, the degrees of the terms should be decreasing as much as possible; for terms with the same (total) degree, the terms must have different degrees for at least one of the variables, and the first variable (alphabetically) whose degree changes must decrease. Finally, 0 should not be the coefficient of any term; the exception to this is that if every coefficient is 0, then we keep the constant term so that we have at least one term.

So a typical polynomial in standard form is 2x3 + 4x2y − 3y3 + x2 − xy + 1/2. Notice that the respective degrees of these terms are 3, 3, 3, 2, 1, and 0; among the degree-3 terms, the degrees on x are 3, 2, and 0. The largest degree of the terms is 3; we call this the degree of the polynomial as a whole. Notice that the degree of a constatn expression is 0, while the degree of a non-constant linear expression is 1.

It may not be obvious now, but every linear expression is equivalent to one in standard form, every monomial is equivalent to one in standard form, and every polynomial is equivalent to one in standard form. We'll see why next.

Combining like terms

Probably the most important application of an identity in Algebra is combining like terms.

Remember that the terms of a polynomial are monomials. Monomials are like (or alike) if they are the same except (possibly) for their coefficients. In other words, they're like if they have the same variables with the same degrees. For example, the monomials 3xy and 5xy are like; they are both xy-terms. However, the monomial 2x2 is unlike these; it is an x2-term. Evenm 2x2 and 2x3 are unlike, because they have different degrees.

Remember that one of the fundamental identities for real numbers says that (a + b)c = ac + bc for any real numbers a, b, and c. If you substitute the coefficients of two terms for a and b and substitute their common factor for c, then this law (going backwards) tells you how to combine those terms. For example, if a = 3, b = −5, and c = x, then I get (3 − 5)x = 3x − 5x; since 3 − 5 = −2, I can combine like terms to get 3x − 5x = −2x. In this way, you can put any polynomial into standard form once it's given as the sum of monomials in standard form. Just rearrange the terms in the correct order and combine the like terms.

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