POLYNOMIALS 

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 : 

4

Solution : 

4x⁰

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

The degree is 0.

Example 3 : 

8y

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.

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. 

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.

Example 13 :

What must be subtracted from 4x⁴ - 2x³ - 6x² + x - 5, so that the result is 2x² + 3x - 2 ?

Solution :

Let the required polynomial be p(x)

4x⁴ - 2x³ - 6x² + x - 5 - p(x) = 2x² + 3x - 2

Solving for p(x), 

p(x) = 4x⁴ - 2x³ - 6x² + x - 5 - 2x² - 3x + 2

= 4x⁴ - 2x³ -  6x²- 2x² + x - 3x + 2 - 5

= 4x⁴ - 2x³ -  8x² - 2x - 3

So, the required polynomial be 4x⁴ - 2x³ -  8x² - 2x - 3.

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