A **polynomial in one variable** is the sum or difference of terms of the form $ax^n$, where $a$ is a real number and $n$ is a nonnegative integer. Polynomials with one, two, or three terms have their own names.

- A
**monomial**is a polynomial expression consisting of one term. An example of a monomial is $3x^5$. - A
**binomial**is a polynomial expression consisting of two terms. An example of a monomial is $4x^3-1$. - A
**trinomial**is a polynomial expression consisting of three terms. An example of a monomial is $x^2-2x-8$.

**degree of a term**, or degree of a monomial, is the exponent of the variable. A constant term has degree zero. A variable with no exponent has degree one. The

**degree of a polynomial**is the greatest degree of a term in the polynomial. In standard form, the degree of the polynomial is the degree of the first term. The

**leading coefficient**of a polynomial is the coefficient of the first term of a polynomial written in standard form, which places the term with the highest degree first.

### Adding and Subtracting Polynomials

**Like terms**are terms that have the same variable (or variables) with the same exponents.

### Multiplying Polynomials

Two common methods for multiplying polynomials include using different types of properties, including the distributive property and properties of exponents.

To use the distributive property for any two polynomials, distribute the first polynomial to each term in the second polynomial. Repeat this process of distribution as needed, depending on the number of terms in the polynomials. The goal is to multiply each term in the first polynomial by each term in the second polynomial.

To use the product of powers property, multiply two powers with the same base by adding the exponents.Second: ${\color{#c42126}{(-4x)}}(x^3)-{\color{#c42126}{(-4x)}}(2x^2)+{\color{#c42126}{(-4x)}}(1)$

Third: ${\color{#c42126}{(7)}}(x^3)-{\color{#c42126}{(7)}}(2x^2)+{\color{#c42126}{(7)}}(1)$

### Factoring Polynomials

**Factoring**is the process of writing a number or algebraic expression as a product. The factored form of a polynomial may help when graphing a function that has the polynomial as its algebraic rule. There are different techniques for factoring polynomials.

### Polynomial Factoring Methods

Method | Description |
---|---|

Greatest common factor (GCF) | When each term has a common factor, divide the greatest common factor from each term. |

Factor by grouping | This method is most commonly used when there are four terms. Group the terms into two pairs. Then factor out the GCF of each pair of terms. The resulting terms have a common factor that can be factored out. |

Factor a trinomial | Many trinomials of the form $ax^2+bx+c$ can be written as the product of two binomials. Find two numbers that are factors of $ac$ and have a sum of $b$. Use these numbers to write the binomial factors or to write a four-term polynomial and continue with factoring by grouping. |

Perfect square trinomial | Some trinomials fit a pattern:
$\begin{gathered}a^2+2ab+b^2=(a+b)^2\\a^2-2ab+b^2=(a-b)^2\end{gathered}$ |

Difference of squares | Some binomials fit a pattern with two squared terms:
$a^2-b^2=(a+b)(a-b)$ |

Sum or difference of cubes | Some binomials fit a pattern with two cubed terms:
$\begin{gathered}a^3+b^3=(a+b)(a^2-ab+b^2)\\a^3-b^3=(a-b)(a^2+ab+b^2)\end{gathered}$ |

Identify whether the binomial fits a pattern.

The binomial is a difference of cubes:### Dividing Polynomials

There are different techniques for dividing polynomials, depending on the polynomials that are given. The combination of monomials and polynomials will determine which technique to use. To divide by a monomial, divide each term of the dividend by the monomial. To divide by a polynomial, start by trying to factor the dividend and divisor.

The quotient of powers property states: To divide two powers with the same base, subtract the exponents.**Synthetic division**is a process of long division of polynomials where only the coefficients and constants are recorded. The divisor must be a linear factor with a coefficient of 1. Before beginning, check that the dividend is written in standard form.

Use the values in the bottom row as coefficients for each term in the quotient.

The degree of the quotient will be one less than the degree of the original dividend. After the constant term, write the remainder as a fraction with the divisor as the denominator.