POLYNOMIAL OPERATIONS

Here's what you'll learn in this
lesson:

Adding and Subtracting

a. Definition of polynomial, term, and
coefficient

b. Evaluating a polynomial

c. The degree of a term and a
polynomial

d. Writing the terms of a polynomial in
descending order

e. Definition of a monomial, binomial,
and trinomial

f. Recognizing like or similar terms

g. Combining like or similar terms

h. Polynomial addition

i. Polynomial subtraction

Multiplying and Dividing

a. Multiplying a monomial by a
monomial

b. Multiplying a monomial by a
polynomial

c. Dividing a monomial by a monomial

d. Dividing a polynomial by a monomial

OVERVIEW

Every day, people use algebra to find unknown quantities. For example, you may be
interested in figuring out how long it would take you to drive across the country. Or, you
may want to know why your checkbook doesn't balance.

To find these unknown quantities, you need to be able to add, subtract, multiply, and
divide polynomials. That’s what you will learn in this lesson.

EXPLAIN

ADDING AND SUBTRACTING

Summary

Identifying Polynomials

A polynomial is a special kind of algebraic expression which may have one or more
variables and one or more terms .

For a polynomial in one variable, x, each term has the form ax^r, where the coefficient, a,
is any real number , and the exponent, r, is a nonnegative integer.

For example:

For a polynomial in two variables , x and y, each term has the form ax^ry^s, where a is any
real number , and r and s are nonnegative integers.

For example:

Polynomials with one, two, or three terms have special names:

A monomial has one term:

A binomial has two terms:

A trinomial has three terms:

An algebraic expression is not a polynomial if any of its terms cannot be written in the
form ax^r.

For example, these algebraic expressions are not polynomials:

Remember, when x≠ 0:
x ⁰= 1
x ¹ = x

The Degree of a Polynomial

The degree of a term of a polynomial is the sum of the exponents of the variables in that
term. The degree of a polynomial is the degree of the term with the highest degree.

For example, to find the degree of the polynomial ; find the degree of
each term:

The degree of the polynomial is the degree of the term with the
highest degree, 7.

Evaluating Polynomials

Sometimes the variables in a polynomial are assigned specific numerical values . In these
cases you can evaluate the polynomial by replacing the variables with the numbers.

To evaluate a polynomial:

1. Replace each variable with its assigned value.

2. Calculate the value of the polynomial .

For example, to evaluate the polynomial when b = 2 and c = 3:


1. Replace b with 2 and c with 3.
2. Calculate.

So, when b = 2 and c = 3, =9.

Adding Polynomials

To add polynomials, combine like terms - terms that have the same variables with the
same exponents.

Here is an example of two like terms:

3x²y and –2x²y

Here is an example of two terms that are
not like terms:

3x²y and 3xy²

To add polynomials:

1. Remove the parentheses .

2. Write like terms next to each other.

3. Combine like terms.

For example, to find:
1. Remove the parentheses.
2. Write like terms next to each other.
3. Combine like terms.

Subtracting Polynomials

To subtract one polynomial from another, add the opposite of the polynomial being subtracted.

To subtract polynomials:

1. Multiply the polynomial being subtracted by –1.

2. Distribute the –1.

3. Simplify.

4. Write like terms next to each other.

5. Combine like terms.

For example, to find:

1. Multiply the second
polynomial by –1.
2. Distribute the –1.
3. Simplify.
4. Write like terms next
to each other.
5. Combine like terms.

To find the opposite of a polynomial,
multiply each term by –1.

When you add the opposite, the result is
the same as changing the sign of each
term in the polynomial being subtracted.

Answers to Sample Problems

b. –2, –2
c. 250, 60, –6 321

b. 3x², 5xy², 3
c. 4x², 7xy², 5

(in any order that like terms are next
to each other)

(in any order)

Sample Problems

1. Evaluate the polynomial when r = 5 and s = –2.

Evaluate:
a. Replace r with 5.
b. Replace s with –2.
c. Calculate.

2. Find:

a. Remove the parentheses.
b. Write like terms next to each other.
c. Combine like terms.

3. Find:

a. Multiply each term in the second polynomial by –1.
b. Distribute the –1.
c. Simplify.
d. Write like terms next to each other.
e. Combine like terms.

MULTIPLYING AND DIVIDING

Summary

Multiplying Monomials

Monomials are easy to multiply because they each have only one term.

To multiply monomials:

1. Rearrange the factors so that the constants are next to each other and the factors with
the same base are next to each other.

2. Multiply.

For example, to find:

1. Rearrange the factors.
2. Multiply.

Multiplying a Monomial by a Polynomial with More Than One Term

When multiplying a monomial by a polynomial with more than one term, you need to
multiply every term in the polynomial by the monomial.

To multiply a monomial by a polynomial with more than one term:

1. Distribute the monomial to each term in the other polynomial.

2. Simplify.

For example, to find:

1. Distribute the monomial.

2. Simplify.

In general, to multiply a monomial by a
polynomial with more than one term:

a(b + c + d) = a· b + a· c + a ·d

Remember:

When
When

The y is on the bottom since the
exponent of y in the denominator is
bigger than the exponent of y in the
numerator .

Dividing Monomials

To divide monomials:

1. Write the division as a fraction.

2. Cancel common numerical factors.

3. Divide factors with the same base by subtracting exponents.

For example, to find

1. Write the division as a fraction.
2. Cancel common numerical factors.
3. Divide factors with the same base.

Dividing a Polynomial with More Than One Term by a Monomial

When you divide a polynomial with more than one term by a monomial, you must divide
each term of the polynomial by the monomial.

Use this rule :

To divide a polynomial with more than one term by a monomial:

1. Write the division as a fraction.
2. Rewrite the fraction using the rule

3. Perform the division on each of the resulting terms.

For example, to find

1. Write the division as a fraction.
2. Rewrite the fraction using the rule
3. Divide each of the resulting terms.

Sample Problems

1. Find:

a. Rearrange the factors so the constants
are next to each other and factors with
the same base are next to each other.

b. Multiply factors with the same
base by adding the exponents.

2. Find: pr ²s (p ²r + pr³s⁶ – 2)

a. Distribute the monomial.

b. Multiply each of the
resulting terms.

3. Find:

a. Write the division as a fraction.

b. Cancel common numerical factors.

c. Divide factors with the same base by
subtracting exponents.

4. Find:

a. Write the division as a fraction.
b. Rewrite the fraction using the rule
c. Divide each term.

Answers to Sample Problems

b. 2, 5, 11, 2

HOMEWORK

Homework Problems

Circle the homework problems assigned to you by the computer, then complete them below.

Explain
Adding and Subtracting

1. Circle the algebraic expression that is a polynomial.

2. Write m beside the monomial, b beside the binomial, and t
beside the trinomial.

3. Given the polynomial

a. write the terms in descending order.

b. find the degree of each term.

c. find the degree of the polynomial.

4. Find:

5. Find:

6. Evaluate when x = 2 and y = 4.

7. Find:

8. Find:

9. Angelina works at a pet store. Today, she is cleaning three
fish tanks. These polynomials describe the volumes of the
tanks:
Tank1:x y²

Tank 2:

Tank 3:

Write a polynomial that describes the total volume of the
three tanks.

Hint: Add the polynomials.

volume = ________

10. Angelina has three fish tanks to clean. These polynomials
describe their volumes.

Tank 1: xy²
Tank 2: x²y – 2y³ + 4xy²+ 3

Tank 3: x²y + 5xy² + 6y³

What is the total volume of the fish tanks if x = 3 feet
and y = 1.5 feet?

volume = ________ cubic feet

11. Find:

12. Find:

Multiplying and Dividing

13. Find:

14. Find:

15. Find:

16. Find:

17. Find:

18. Find:

19. Write:as a fraction and simplify .

20. Write:as a fraction and simplify.

21. Find:

22. Tony is an algebra student. This is how he answered a
question on a test:

Is his answer right or wrong? Why? Circle the most
appropriate response.

The answer is right.

The answer is wrong. Tony divided the exponents rather
than adding them. The correct answer is

The answer is wrong. The terms need to be ordered by
degree. The correct answer is

The answer is wrong. Tony divided the exponents rather
than subtracting them. The correct answer is

The answer is wrong. Tony shouldn't have canceled the
numerical coefficients. The correct answer is

23.Find:

24.Find:

APPLY

Practice Problems

Here are some additional practice problems for you to try.

Adding and Subtracting

1. Circle the algebraic expressions below that are polynomials.

2. Circle the algebraic expressions below that are polynomials.

3. Identify each polynomial below as a monomial, a binomial, or
a trinomial.

4. Identify each polynomial below as a monomial, a binomial, or
a trinomial.

5. Find the degree of the polynomial

6. Find the degree of the polynomial

7. Find the degree of the polynomial

8. Evaluate when x = –1.

9. Evaluate when x = –2.

10. Evaluate when x = 3.

11. Evaluate when x = 2 and y = –3.

12. Evaluate when m 4 and n –2.

13. Evaluate when u 2 and v –4.