Polynomials

There are many different types of algebraic expressions, but most of the expressions in this class are a particular type, called a polynomial. A polynomial is any expression built out of constants and variables using the operations of addition and multiplication, or anything equivalent to this. (You don't have to use all of these ingredients in any given polynomial.) Each polynomial has a standard form; usually the first step in dealing with any expression is to put it in standard form.

Standard forms

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.

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.

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 (no variables). Then there is one factor for each variable, which consists of that variable raised to the power of a whole number. This whole number 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 has 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.

When writing out a polynomial, 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 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.

Operations on polynomials

Two terms are alike (or 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.

This also 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.

Similarly, two factors are alike (or 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.

This also 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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