POLYNOMIALS

A polynomial is a monomial or a sum or difference of monomials. The degree of a polynomial is the degree of the term with the greatest degree.

A monomial is a number, a variable, or a product of numbers and variables with whole number exponents.

5, x, -7xy, 0.5x⁴

-0.3x^-2, 4x - y, 2/x³

The degree of a monomial is the sum of the exponents of the variables. A constant has degree 0.

Finding the Degree of a Monomial

Find the degree of each monomial.

Example 1 :

-2a²b⁴

Solution :

Add the exponents of the variables : 2 + 4 = 6.

The degree is 6.

Example 2 :

Solution :

4x⁰

There is no variable, but you can write 4 as 4x⁰ .

The degree is 0.

Example 3 :

Solution :

8y¹

A variable written without an exponent has exponent 1.

The degree is 1.

Finding the Degree of a Polynomial

Find the degree of each polynomial.

Example 4 :

4x - 18x⁵

Solution :

Find the degree of each term.

4x : degree 1

-18x⁵ : degree 5

The degree of the polynomial is the greatest degree, 5.

Example 5 :

0.5x²y + 0.25xy + 0.75

Solution :

Find the degree of each term.

0.5x²y = degree 3

0.25xy = degree 2

0.75 = degree 0

The degree of the polynomial is the greatest degree, 3.

Example 6 :

6x⁴ + 9x² - x + 3

Solution :

Find the degree of each term.

6x⁴ = degree 4

9x² = degree 2

-x = degree 1

3 = degree 0

The degree of the polynomial is the greatest degree, 4.

The terms of a polynomial may be written in any order. However, polynomials that contain only one variable are usually written in standard form.

The standard form of a polynomial that contains one variable is written with the terms in order from greatest degree to least degree. When written in standard form, the coefficient of the first term is called the leading coefficient.

Writing Polynomials in Standard Form

Write each polynomial in standard form. Then give the leading coefficient.

Example 7 :

20x - 4x³ + 2 - x²

Solution :

Find the degree of each term.

20x = degree 1

-4x³ = degree 3

2 = degree 0

-x² = degree 2

Arrange them in descending order :

-4x³ - x²+ 20x + 2

The standard form is (-4x³ - x²+ 20x + 2) and the leading coefficient is -4.

Example 8 :

y³ + y⁵+ 4y

Solution :

Find the degree of each term.

y³ = degree 3

y⁵ = degree 5

4y = degree 1

Arrange them in descending order :

y⁵ + y³+ 4y

The standard form is (y⁵ + y³+ 4y) and the leading coefficient is 1.

Some polynomials have special names based on their degree and the number of terms they have.

Degree

6 or more

Name

Constant

Linear

Quadratic

Cubic

Quartic

Quintic

6th degree, 7th degree, and so on

Terms

4 or more

Name

Monomial

Binomial

Trinomial

Polynomial

Classifying Polynomials

Classify each polynomial according to its degree and number of terms.

Example 9 :

5x - 6

Solution :

Degree : 1

Terms : 2

(5x - 6) is a linear binomial.

Example 10 :

y² + y + 4

Solution :

Degree : 2

Terms : 3

(y² + y + 4) is a quadratic trinomial.

Example 11 :

5x⁷ + 8x² - x + 3

Solution :

Degree : 7

Terms : 4

(5x⁷ + 8x² - x + 3) is a 7^th-degree trinomial.

Physics Application

Example 12 :

A firework is launched from a platform 6 feet above the ground at a speed of 200 feet per second. The firework has a 5-second fuse. The height of the firework in feet is given by the polynomial -16t² + 200t + 6, where t is the time in seconds. How high will the firework be when it explodes?

Solution :

Substitute the time for t to find the firework’s height.

-16t² + 200t + 6

The time is 5 seconds.

= -16(5)² + 200(5) + 6

Evaluate the polynomial by using the order of operations.

= -16(25) + 1000 + 6

= -400 + 1000 + 6

= 606

When the firework explodes, it will be 606 feet above the ground.