Exponents, Polynomials, and Polynomial Functions · Know the basic definitions for polynomials....

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Exponents, Polynomials, and Polynomial Functions

R.1 Fractions

1.  Know the basic definitions for polynomials. 2.  Add and subtract polynomials.

Objectives

5.2 Adding and Subtracting Polynomials

Know the basic definitions for polynomials.

Polynomials are fundamental in algebra. Recall that a term is a number, a variable, or the product or quotient of a number of one or more variables raised to powers.

Examples of terms include:

Coefficients are written in blue.

Recall that any combination of variables or constants (numerical values) joined by the basic operations of addition, subtraction, multiplication, and division (except by 0), or raising to powers or taking roots is called an algebraic expression. Polynomials are the simplest kind of algebraic expression.

A polynomial containing only the variable x is called a polynomial in x.

A polynomial in one variable is written in descending powers of the variable if the exponents on the variable decrease from left to right.

The powers of x are decreasing from left to right. We can think of this polynomial as

The Degree of a Polynomial

The degree of a nonzero term with only one variable is the exponent on the variable. The number 0 has no degree. The degree of a polynomial is the highest degree of any of its nonzero terms.

Some Common Polynomials

Polynomials having a specific number of terms are commonly given special names.

Trinomial = a polynomial with three terms

Binomial = a polynomial with two terms

Monomial = a polynomial with one term

Classifying Polynomials Example 2 Identify each polynomial as a monomial, a binomial, a trinomial, or none of these. Also, give the degree. a. b. c.

2 5 1x x− + +

4 2 6 5p p p− − −

Trinomial; degree 2.

Monomial; degree 5 (1 + 4 = 5)

None of these; degree 4.

Add and Subtract Polynomials

Addition of polynomials is just a matter of adding up like terms. For example, consider the following polynomials:

We can use the associative and commutative properties to rearrange the terms

and then we add the like terms.

Adding Polynomials

Add: Vertical Solution

Example 4

Subtracting Polynomials

Subtract.

Example 5

We can subtract these polynomials vertically by writing the first polynomial above the second, lining up like terms in columns.

Change all the signs in the second polynomial, and add.

Example 5

Exponents, Polynomials, and Polynomial Functions · Know the basic definitions for polynomials....

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