Polynomials

Definitions

Definition:  A polynomial in terms of x is an expression that can be expressed as a sum with the form:

Where all powers, n, n-1,…,1,0 are  non-negative integers and all coefficients an , an-1 , … , a1 , a0 are real numbers.

The highest power of x that has a non-zero coefficient is called the order of the polynomial.

The non-zero coefficient an of the highest power is called the leading coefficient of the polynomial.

 

Examples of Polynomials:

Example

Leading Coefficient

Order

3

4

1

8

-5

2

8

8

0

7

1

 

Examples of expressions that are not polynomials:

 Example 1)           

       The term  has a negative exponent, contradicting the definition of a polynomial

Example 2)                  

 

       After the expression has been completely simplified, there are expressions with x in the denominator of a fraction.  Correspondingly, it cannot be expressed as a sum with the form that the definition requires and is not a polynomial.

Example 3)               

     The power in the term  is not an integer, contradicting the definition of a polynomial.

 

Click here to practice definitions associated with polynomials

 

Operations with Polynomials:

Sums: To add two polynomails together, we follow the steps below.

Step 1:  In each polynomial rewrite subtraction as addition.

Step 2.  Group terms that have the same power of x (similar terms)

Step 3.  Add similar terms.

Example: If  and then find the sum 

Step 1:  Rewrite addition as subtraction

,  

Step 2:  Group Similar Terms

  

Step 3:  Add similar terms

 = 

Solution:

 

 

 

Differences: To subtract one polynomial from another, we follow the steps below.

Step 1:  In each polynomial rewrite subtraction as addition and add 0 coefficients to missing powers of x.

Step 2.  Group terms that have the same power of x (similar terms) placing the term from the polynomial to be subtracted second.

Step 3.  Subtract the second similar term from the first.

Example: If  and then find  

Step 1:  Rewrite addition as subtraction

,  

Step 2:  Group Similar Terms

  

Step 3:  Subtract similar terms

 = 

Solution:

 

Click here to practice adding and subtracting polynomials