A **polynomial** consists of two or more terms. For example, *x* + *y*, *y*^{2} – *x*^{2}, and *x*^{2} + 3 *x* + 5 *y*^{2} are all polynomials. A **binomial** is a polynomial that consists of exactly two terms. For example, *x* + *y* is a binomial. A **trinomial** is a polynomial that consists of exactly three terms. For example, *y*^{2} + 9 *y* + 8 is a trinomial.

Polynomials usually are arranged in one of two ways. **Ascending order** is basically when the power of a term increases for each succeeding term. For example, *x* + *x*^{2} + *x*^{3} or 5 *x* + 2 *x*^{2} – 3 *x*^{3} + *x*^{5} are arranged in ascending order. **Descending order** is basically when the power of a term decreases for each succeeding term. For example, *x*^{3} + *x*^{2} + *x* or 2 *x*^{4} + 3 *x*^{2} + 7 *x* are arranged in descending order. Descending order is more commonly used.

Adding and subtracting polynomials

To *add* or *subtract polynomials*, just arrange *like terms* in columns and then add or subtract. (Or simply add or subtract like terms when rearrangement is not necessary.)

Example 1

Do the indicated arithmetic.

- Add the polynomials.
- Subtract the polynomials.

Multiplying polynomials

To *multiply polynomials,* multiply each term in one polynomial by each term in the other polynomial. Then simplify if necessary.

Example 2

Multiply.

Or you may want to use the “ **F.O.I.L.**” method with *binomials.* **F.O.I.L.** means **F**irst terms, **O**utside terms, **I**nside terms, **L**ast terms. Then simplify if necessary.

Example 3

Multiply.

(3 *x* + *a*)(2 *x* – 2 *a*) =

Multiply *first* terms from each quantity.

Now *outside* terms.

Now *inside* terms.

Finally *last* terms.

Now simplify.

6 *x*^{2} – 6 *ax* + 2 *ax* – 2 *a*^{2} = 6 *x*^{2} – 4 *ax* – 2 *a*^{2}

Example 4

Multiply.

This operation also can be done using the distributive property.

Dividing polynomials by monomials

To *divide a polynomial by a monomial,* just divide each term in the polynomial by the monomial.

Example 5

Divide.

Dividing polynomials by polynomials

To *divide a polynomial by a polynomial,* make sure both are in descending order; then use long division. ( *Remember:* Divide by the first term, multiply, subtract, bring down.)

Example 6

Divide 4 *a*^{2} + 18 *a* + 8 by *a* + 4.

Example 7

Divide.

- First change to descending order:
*x*^{2}+ 2*x*+ 1. Then divide. - Note: When terms are missing, be sure to leave proper room between terms.
- This answer can be rewritten as