### Learning Outcomes

- Define, evaluate, and simplify single variable polynomials

In the example on the previous page, we saw how combining the formulas for different shapes provides a way to accurately predict the amount of paint needed for a construction project. The result was a **polynomial**.

A polynomial function is a function consisting of the sum or difference of terms in which each term is a real number, a variable, or the product of a real number and variable(s) with a non-negative integer exponents. Non negative integers are [latex]0, 1, 2, 3, 4[/latex], …

You may see a resemblance between expressions and polynomials which we have been studying in this course. Polynomials are a special sub-group of mathematical expressions and equations.

The following table is intended to help you tell the difference between what is a polynomial and what is not.

IS a Polynomial | Is NOT a Polynomial | Reason |

[latex]2x^2-\frac{1}{2}x -9[/latex] | [latex]\frac{2}{x^{2}}+x[/latex] | Polynomials only have variables in the numerator |

[latex]\frac{y}{4}-y^3[/latex] | [latex]\frac{2}{y}+4[/latex] | Polynomials only have variables in the numerator |

[latex]\sqrt{12}\left(a\right)+9[/latex] | [latex]\sqrt{a}+7[/latex] | Roots are equivalent to rational exponents, and polynomials only have integer exponents on variables |

The basic building block of a polynomial is a **monomial**. A monomial is one term and can be a number, a variable, or the product of a number and variable(s) with an exponent. The number part of the term is called the **coefficient**.

A polynomial containing two terms, such as [latex]2x - 9[/latex], is called a **binomial**. A polynomial containing three terms, such as [latex]-3{x}^{2}+8x - 7[/latex], is called a **trinomial**.

We can find the **degree** of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the **leading term** because it is usually written first. The coefficient of the leading term is called the **leading coefficient**. When a polynomial is written so that the powers on a specified variable are descending, we say that it is in standard form. It is important to note that polynomials only have integer exponents.

### How to: Given a polynomial expression, identify the degree and leading coefficient

- Find the highest power of
*x*to determine the degree. - Identify the term containing the highest power of
*x*to find the leading term. - Identify the coefficient of the leading term.

### Example

For the following polynomials, identify the degree, the leading term, and the leading coefficient.

- [latex]3+2{x}^{2}-4{x}^{3}[/latex]
- [latex]5{t}^{5}-2{t}^{3}+7t[/latex]
- [latex]6p-{p}^{3}-2[/latex]

In the following video, we will identify the terms, leading coefficient, and degree of a polynomial.

The table below illustrates some examples of monomials, binomials, trinomials, and other polynomials. They are all written in standard form.

Monomials |
Binomials |
Trinomials |
Other Polynomials |

[latex]15[/latex] | [latex]3y+13[/latex] | [latex]x^{3}-x^{2}+1[/latex] | [latex]5x^{4}+3x^{3}-6x^{2}+2x[/latex] |

[latex] \displaystyle \frac{1}{2}x[/latex] | [latex]4p-7[/latex] | [latex]3x^{2}+2x-9[/latex] | [latex]\frac{1}{3}x^{5}-2x^{4}+\frac{2}{9}x^{3}-x^{2}+4x-\frac{5}{6}[/latex] |

[latex]-4y^{3}[/latex] | [latex]3x^{2}+\frac{5}{8}x[/latex] | [latex]3y^{3}+y^{2}-2[/latex] | [latex]3t^{3}-3t^{2}-3t-3[/latex] |

[latex]16n^{4}[/latex] | [latex]14y^{3}+3y[/latex] | [latex]a^{7}+2a^{5}-3a^{3}[/latex] | [latex]q^{7}+2q^{5}-3q^{3}+q[/latex] |

When the coefficient of a polynomial term is [latex]0[/latex], you usually do not write the term at all (because [latex]0[/latex] times anything is [latex]0[/latex], and adding [latex]0[/latex] does not change the value). The last binomial above could be written as a trinomial, [latex]14y^{3}+0y^{2}+3y[/latex].

A term without a variable is called a **constant **term, and the degree of that term is [latex]0[/latex]. For example, [latex]13[/latex] is the constant term in [latex]3y+13[/latex]. You would usually say that [latex]14y^{3}+3y[/latex] has no constant term or that the constant term is [latex]0[/latex].

## Evaluate a Polynomial

You can evaluate polynomials just as you have been evaluating expressions all along. To evaluate an expression for a value of the variable, you substitute the value for the variable *every time* it appears. Then use the order of operations to find the resulting value for the expression.

### Example

Evaluate [latex]3x^{2}-2x+1[/latex] for [latex]x=-1[/latex].

### Example

Evaluate [latex] \displaystyle -\frac{2}{3}p^{4}+2p^{3}-p[/latex] for [latex]p = 3[/latex].

In the following video, we show more examples of evaluating polynomials for given values of the variable.