Polynomials

There are many different types of algebraic expressions, but most of the expressions in this class are a particular type: a polynomial.

A polynomial is an algebraic expression built out of constants and variables using the operations of addition and multiplication, or anything equivalent to this. Thus, anything built using addition, subtraction, opposites, multiplication, division by nonzero constants, and raising to powers of whole constants is a polynomial. (This is because subtraction is adding an opposite, you can take an opposite by multiplying by −1, dividing by a nonzero constant is multiplying by its reciprocal, raising to the power of zero is equivalent to the constant 1, and raising to the power of a nonzero whole number is equivalent to repeated multiplication.) However, it is forbidden in a polynomial to divide by zero or a variable expression (or to take the reciprocal of zero or a variable expression), to raise to the power of a negative or fractional number or of a variable, or to take an absolute value.

Usually the first step in dealing with any polynomial is to simplify it to standard form; for example, x + 5 is considered standard while 5 + x is not, so 5 + x simplifies to x + 5. Exactly what counts as the standard form of an expression will depend on the context that you're working in, so ‘simplify’ doesn't always mean to make things simpler (but it usually at least doesn't make them any more complicated).

Every polynomial is a sum of terms; you add up the terms to get the entire expression. Here are a couple of examples:

The terms of 2x² + 3x + 4 are 2x², 3x, and 4;
The terms of x³ − 2x² + 3 are x³, −2x², and 3.

The last example shows that, when analysing polynomials, you want to think of subtraction as a form of addition. Here, we think of subtracting 2x² as adding −2x². This is one reason why I began these notes by saying that polynomials are built from variables and constants using only addition and multiplication; every use of subtraction should be thought of as really a use of addition.

Monomials

Each term of a polynomial is a monomial; a monomial is any expression built out of constants and variables using only the operation of multiplication, or anything equivalent to this. Anything built using opposites, multiplication, division by nonzero constants, and raising to powers of whole constants is a monomial, but addition or subtraction is forbidden.

Every monomial is a product of factors; you multiply the factors to get the entire term. The first factor is the coefficient, which is always a constant. Then there is one factor for each variable, which consists of that variable raised to the power of a whole constant. This whole constant is the degree of the monomial on that variable. So in the end, a monomial is given by several numbers: the coefficient and the degrees on all of the variables.

Think about these special cases before we look at some examples:

If there is no constant factor, then the coefficient is 1.
If there is no exponent on a variable, then the degree on that variable is 1.
If a variable does not appear, then the degree on that variable is 0.

Now here are some examples:

The coefficient of 2x² is 2, and its degree on x is 2.
The coefficient of 3x is 3, and its degree on x is 1. (Remember that x¹ = x, so we can think of 3x as 3x¹.)
The coefficient of 4 is 4, and its degree on x is 0. (Remember that x⁰ = 1, multiplication by which does nothing, so we can think of 4 as 4x⁰.)
The coefficient of x³ is 1, and its degree on x is 3. (Remember that multiplication by 1 does nothing, so we can think of x³ as 1x³.)
The coefficient of −2x² is −2, and its degree on x is 2.
The coefficient of 3 is 3, and its degree on x is 0.

Now if you look back at the first two examples of polynomials, we see that each of these polynomials has three terms, and each of these terms is a monomial with a coefficient and a degree on x.

So far, I've only talked about degrees on specific variables. 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 the degree on x is 2 and the degree on y is 3 (and 2 + 3 = 5).
The degree of 4xy³ is 4, since the degree on x is 1 (and 1 + 3 = 4).
The degree of the constant monomial 7 is 0, since there is nothing to add; the degree on every variable is 0.

Other polynomials

When writing out a polynomial in standard form, we always write the terms in order of decreasing degree. The degree of the polynomial as a whole is the largest degree of any of its nonzero terms. Returning to our original examples:

The degrees of the terms of 2x² + 3x + 4 are 2, 1, and 0, so the degree of this polynomial is 2.
The degree of x³ − 2x² + 3 is 3, since the degrees of its terms are 3, 2, and 0.

A polynomial whose degree is zero is equivalent to a constant. (Since the constant 0 has no nonzero terms, it technically does not have a degree at all; sometimes people say that its degree is −1, or even −∞, meaning negative infinity, but I will say that the degree is undefined.)

A linear expression is a polynomial whose degree is at most 1; in other words, either it is constant or its degree is exactly 1. A linear expression can be built out of constants and variables using only addition and multiplication by constants; any expression built from addition, subtraction, opposites, multiplication by constants, and division by constants is linear, but multiplication by a variable expression or raising to a power (other than 0 or 1) is forbidden. For example, neither 2x² + 3x + 4 nor x³ − 2x² + 3, but 2x + 3y + 4 is linear, as is a − 2b + 3. Most of the first half of this course will be about linear expressions.

A polynomial is quadratic if its degree is 2, or cubic if its degree is 3. Thus, 2x² + 3x + 4 is quadratic, while x³ − 2x² + 3 is cubic. There are additional words (now based on Latin words for the numbers) for polynomials of degree 4, 5, or more, but these are not used much. (Even ‘cubic’ is not used in the textbook.) There doesn't seem to be a word for a polynomial whose degree is 1 other than ‘nonconstant linear’, and there doesn't seem to be a word for a polynomial whose degree is 0 other than ‘nonzero constant’.

Notice that a monomial is simply a polynomial with 1 term. Similarly, a binomial is a polynomial with 2 terms, a trinomial is a polynomial with 3 terms, and a tetranomial is a polynomial with 4 terms. For example, both 2x² + 3x + 4 and x³ − 2x² + 3 are trinomials. There are additional words (based on Greek words for the numbers) for polynomials with 5, 6, or more terms, but these are not used much. (Even ‘tetranomial’ is not used in the textbook.)

Addition and subtraction

Two terms are alike (or are like terms) if they are the same except for their coefficients. For example:

In 2x + 3x, 2x and 3x are alike, since they are both x-terms;
In 2x + 2y, 2x and 2y are not alike, since one is an x-term but the other is a y-term;
In 2x + 2x², 2x and 2x² are also not alike, since one is an x-term but the other is an x²-term.

You can combine like terms into a single term by adding the coefficients; for example:

2x + 3x = 5x.

A polynomial is in standard form if it is written as a sum of terms, each of its terms is a monomial in standard form (see the next section), no two of its terms are alike, none of its terms has a coefficient of 0 (unless the entire polynomial is the constant 0), and the terms are in order of decreasing degree. (If there is more than one variable, then there may be multiple terms with the same degree; then we usually put them in alphabetical order as much as possible.)

Combining like terms defines addition of polynomials; for example:

To add 2x² + 3x + 4 and x³ − 2x² + 3, we get x³ + 0x² + 3x + 7, which is x³ + 3x + 7.

You can also take the opposite of a polynomial by taking the opposite of each coefficient in each term; for example:

The opposite of 2x² + 3x + 4 is −2x² − 3x − 4;
The opposite of x³ − 2x² + 3 is −x³ + 2x² − 3.

This now defines subtraction of polynomials; for example:

To subtract 2x² + 3x + 4 from x³ − 2x² + 3, we add −2x² − 3x − 4 to x³ − 2x² + 3 to get x³ − 4x² − 3x − 1.

Multiplication

Similar to like terms, two factors are alike (or are like factors) if they are both coefficients or involve the same variable. For example:

In x²x³, x² and x³ are alike, since they are both x-factors;
In x²y², x² and y² are not alike, since one is an x-factor but the other is a y-factor;
In 2x², 2 and x² are also not alike, since one is a coefficient but the other is an x-factor.

You can combine like factors into a single factor by adding the degrees; for example:

x²x³ = x⁵.

A monomial is in standard form if it is written as a sum of factors, no two of its factors are alike, none of its factors has a degree of 0, 1 is not a factor (unless the entire monomial is the constant 1), a factor of −1 is written as a minus sign (unless the entire monomial is the constant −1), and no other factors appear if 0 is a factor. (The order of the variable factors is not specified, but we usually put them in alphabetical order.)

Combining like factors defines multiplication of monomials; for example:

To multiply 2x²y³ and x³y², we get 2x⁵y⁵.

Finally, to define multiplication of polynomials, we multiply each term by each term. For example:

To multiply 2x² + 3x + 4 and x³ − 2x² + 3, we get:
- (2x²)(x³) + (2x²)(−2x²) + (2x²)(3) + (3x)(x³) + (3x)(−2x²) + (3x)(3) + (4)(x³) + (4)(−2x²) + (4)(3), which is
- 2x⁵ − 4x⁴ + 6x² + 3x⁴ − 6x³ + 9x + 4x³ − 8x² + 12, which is
- 2x⁵ − 4x⁴ + 3x⁴ − 6x³ + 4x³ + 6x² − 8x² + 9x + 12, which is
- 2x⁵ − x⁴ − 2x³ − 2x² + 9x + 12.

This example also shows the importance of putting the terms of a polynomial in order of decreasing degree.

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