# Read: Define and Evaluate Polynomials

### Learning Objectives

- Identify the degree and leading coefficient of a polynomial
- Evaluate a polynomial for given values

**polynomial**.

A polynomial function is a function consisting of sum or difference of terms in which each term is a real number, a variable, or the product of a real number and variables with an non-negative integer exponents. Non negative integers are

$0, 1, 2, 3, 4$

, ...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 | Because |

$2x^2-\frac{1}{2}x -9$ |
$\frac{2}{x^{2}}+x$ |
Polynomials only have variables in the numerator |

$\frac{y}{4}-y^3$ |
$\frac{2}{y}+4$ |
Polynomials only have variables in the numerator |

$\sqrt{12}\left(a\right)+9$ |
$\sqrt{a}+7$ |
Roots are equivalent to rational exponents, and polynomials only have integer exponents |

**monomial**. A monomial is one term and can be a number, a variable, or the product of a number and variables with an exponent. The number part of the term is called the

**coefficient**.

A polynomial containing two terms, such as

$2x - 9$

, is called a **binomial**. A polynomial containing three terms, such as

$-3{x}^{2}+8x - 7$

, 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 are descending, we say that it is in standard form. It is important to note that polynomials only have integer exponents.

#### 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.- $3+2{x}^{2}-4{x}^{3}$
- $5{t}^{5}-2{t}^{3}+7t$
- $6p-{p}^{3}-2$

Show Answer

- The highest power of
*x*is$3$, so the degree is$3$. The leading term is the term containing that degree,$-4{x}^{3}$. The leading coefficient is the coefficient of that term,$-4$. - The highest power of
*t*is$5$, so the degree is$5$. The leading term is the term containing that degree,$5{t}^{5}$. The leading coefficient is the coefficient of that term,$5$. - The highest power of
*p*is$3$, so the degree is$3$. The leading term is the term containing that degree,$-{p}^{3}$, The leading coefficient is the coefficient of that term,$-1$.

In the following video example, 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 |

$15$ |
$3y+13$ |
$x^{3}-x^{2}+1$ |
$5x^{4}+3x^{3}-6x^{2}+2x$ |

$\displaystyle \frac{1}{2}x$ |
$4p-7$ |
$3x^{2}+2x-9$ |
$\frac{1}{3}x^{5}-2x^{4}+\frac{2}{9}x^{3}-x^{2}+4x-\frac{5}{6}$ |

$-4y^{3}$ |
$3x^{2}+\frac{5}{8}x$ |
$3y^{3}+y^{2}-2$ |
$3t^{3}-3t^{2}-3t-3$ |

$16n^{4}$ |
$14y^{3}+3y$ |
$a^{7}+2a^{5}-3a^{3}$ |
$q^{7}+2q^{5}-3q^{3}+q$ |

$0$

, you usually do not write the term at all (because $0$

times anything is $0$

, and adding $0$

doesn’t change the value). The last binomial above could be written as a trinomial, $14y^{3}+0y^{2}+3y$

.A term without a variable is called a

**constant**term, and the degree of that term is

$0$

. For example $13$

is the constant term in $3y+13$

. You would usually say that $14y^{3}+3y$

has no constant term or that the constant term is $0$

.## 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$3x^{2}-2x+1$

for $x=-1$

.Show Solution

Substitute

$-1$

for each *x*in the polynomial.$3\left(-1\right)^{2}-2\left(-1\right)+1$

Following the order of operations, evaluate exponents first.$3\left(1\right)-2\left(-1\right)+1$

Multiply $3$

times $1$

, and then multiply $-2$

times $-1$

.$3+\left(-2\right)\left(-1\right)+1$

Change the subtraction to addition of the opposite.$3+2+1$

Find the sum.#### Answer

$3x^{2}-2x+1=6$

, for $x=-1$

### Example

Evaluate$\displaystyle -\frac{2}{3}p^{4}+2^{3}-p$

for $p = 3$

.Show Solution

Substitute

$3$

for each *p*in the polynomial.$\displaystyle -\frac{2}{3}\left(3\right)^{4}+2\left(3\right)^{3}-3$

Following the order of operations, evaluate exponents first and then multiply.$\displaystyle -\frac{2}{3}\left(81\right)+2\left(27\right)-3$

Add and then subtract to get $-3$

.$-54 + 54 – 3$

#### Answer

$\displaystyle -\frac{2}{3}p^{4}+2p^{3}-p=-3$

, for $p = 3$

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