Section

Concepts

2.1 Addition and Subtraction of Polynomials

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1. Introduction to Polynomials2. Addition of Polynomials3. Subtraction of Polynomials

Section

Concepts

2.1 Addition and Subtraction of Polynomials

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Any Homework Questions?

Section 2.1 Addition and Subtraction of Polynomials

1. Introduction to Polynomials

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One commonly used algebraic expression is called a polynomial. A polynomial in one variable, x, is defined as a single term or a sum of terms of the formwhere a is a real number and the exponent, n, is a nonnegative integer. For each term, a is called the coefficient, and n is called the degree of the term.

Section 2.1 Addition and Subtraction of Polynomials

1. Introduction to Polynomials

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A polynomial which contains one term is categorized as amonomial.A polynomial which contains two terms is categorized as a binomial.A polynomial which contains three terms is categorized asa trinomial.An expression which contains four or more terms is categorized a polynomial.

Section 2.1 Addition and Subtraction of Polynomials

1. Introduction to Polynomials

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The degree of a polynomial is the highest power of all of its terms. Thus, when written in descending order, the leading term determines the degree of the polynomial.

Section 2.1 Addition and Subtraction of Polynomials

1. Introduction to Polynomials

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Example 1 Identifying the Parts of a Polynomial

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Given:

a. List the terms of the polynomial, and state the coefficient and degree of each term.

b. Write the polynomial in descending order.c. State the degree of the polynomial and the leading

coefficient.

ExampleSolution:

1 Identifying the Parts of a Polynomial

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a. term:

term:

term:

term:

coefficient:coefficient:

coefficient:

coefficient:

degree:

degree:

degree:

degree:

4.5a

1.6

b.

c. The degree of the polynomial is The leading coefficient is

Write the polynomial in descending order.

Section 2.1 Addition and Subtraction of Polynomials

1. Introduction to Polynomials

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Polynomials may have more than one variable. In such a case, the degree of a term is the sum of the exponents of the variables contained in the term.The following polynomial has a degree of 11 because thehighest degree of its terms is 11.

Section 2.1 Addition and Subtraction of Polynomials

2. Addition of Polynomials

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Recall that two terms are like terms if they each have the same variables, and the corresponding variables are raised to the same powers.

Section 2.1 Addition and Subtraction of Polynomials

2. Addition of Polynomials

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Recall that the distributive property is used to add or subtract like terms. For example,

We can shorten the distributive process by adding coefficients of like terms.

Example 2 Adding Polynomials

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Add the polynomials.

a.

b.

Example 3 Adding Polynomials

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Add the polynomials.

a.

b.

3 2 24 3 5 7x x x x

3 2 32 6 2 3 6 8c c c c c

TIP:

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Polynomials can also be added by combining like terms in columns.

Place holders such as 0 and 0c may be used to help line up like terms.

Add the polynomials

3 2 3 24 7 5 4 3c c c c c

3 2

3 2

4 7 5 0

4 0 3

c c c

c c c

___________________

Example 4 Adding Polynomials

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Add the polynomials.

a.

b. 3 2 6 5a b c a b c

Section 2.1 Addition and Subtraction of Polynomials

3. Subtraction of Polynomials

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Subtraction of two polynomials requires us to find the opposite of the polynomial being subtracted. To find the opposite of a polynomial, take the opposite of each term. This is equivalent to multiplying the polynomial by

ExampleSolution:

5 Finding the Opposite of a Polynomial

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Expression

Opposite

Simplified Form

a. b. c.

TIP:

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Notice that the sign of each term is changedwhen finding the opposite of a polynomial.

DEFINITION Subtraction of Polynomials

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If A and B are polynomials, then

Example 6 Subtracting Polynomials

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Subtract the polynomials.

a.

b.

3 8 2 6x x

TIP:

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Two polynomials can also be subtracted in columns by adding the opposite of the second polynomial to the first polynomial. Place holders (shown in red) may be used to help line up like terms.

The difference of the polynomials is

4 2 24 5 3 11 4 6p p p p

Example 7 Subtracting Polynomials

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Subtract the polynomials

a.

b.

2 23 2 1 1 4 3

8 5 7 8 5 7x x x x

Find the difference of and 25 3 21y y 24 5 23y y

Example 8 Subtracting Polynomials

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Perform the indicated operations

a. 2 2 22 2 4 3 5 2y y y y y y

Example2.5

Addition and Subtraction of Polynomials

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1. Introduction to Polynomials2. Addition of Polynomials3. Subtraction of Polynomials

Example

You Try:

Add the polynomials. 2 24 3 6 4 3t t t t

2 2 2 22 5 4 3 2a b ab ab a b ab ab Subtract the polynomials

Example

You Try:

Perform the indicated operation3.

2 2 25 2 3 3 3 2 8x x x x x x