Chapter 9 Section 1 Polynomials

Vocabulary Polynomial: The sum of one or more monomials is called a polynomial. Monomial: A monomial is a number, a variable, or a product of numbers and variables that have only positive exponents. Binomial: A polynomial with two terms is a binomial. Trinomial: A polynomial with three terms is a trinomial.

What You’ll Learn You’ll learn to identify and classify polynomials and find their degree.

Why it is important One example of why it is important is medicine. Doctors can use polynomials to study the heart.

A cube is a solid figure in which all the faces are square. Suppose you wanted to paint the cube shown below. You would need to find the surface area of the cube to determine how much paint to buy.

The area of each face of the cube is x ∙ x or x 2. There are six faces to paint. x 2 + x 2 + x 2 + x 2 + x 2 + x 2 = 6x 2 So, the surface area of the cube is 6x 2 square feet. x ft.

The expression 6x 2 is called a monomial. A monomial is a number, a variable, or a product of numbers and variables that have only positive exponents. A monomial cannot have a variable as an exponent. MonomialsNot Monomials -4A number2x2x Has a variable as an exponent YA variablex 2 + 3Includes addition a2a2 The product of variables5a -2 Includes a negative exponent ½ x 2 yThe product of numbers and variables 3x3x Includes division

Example 1 Determine whether each expression is a monomial. Explain why or why not. -6ab -6ab is a monomial. It is the product of a number and variables.

Example 2 Determine whether each expression is a monomial. Explain why or why not. m m 2 – 4 is not a monomial, because it includes subtraction.

Your Turn Determine whether each expression is a monomial. Explain why or why not is a monomial, because it is a number.

Your Turn Determine whether each expression is a monomial. Explain why or why not. 5z -3 5z -3 is not a monomial, because it includes a negative exponent.

Your Turn Determine whether each expression is a monomial. Explain why or why not. 6 x This is not a monomial, because it includes division.

Your Turn Determine whether each expression is a monomial. Explain why or why not. x2x2 x 2 is a monomial, because it is a product of variables.

The sum of one or more monomials is called a polynomial. For example, x 3 + x 2 + x + 2 is a polynomial. The terms of the polynomial are x 3, x 2, x, and 2.

Special names are given to polynomials with two or three terms. A polynomial with two terms is a binomial. A polynomial with three terms is a trinomial. Here are some examples. BinomialTrinomial x + 2a + b + c 5c - 4x 2 + 5x - 7 4w 2 - w3a 2 + 5ab + 2b 2

Example 3 State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial. 2m – 7 The expression 2m – 7 can be written as 2m + (-7). So, it is a polynomial. Since it can be written as the sum of two monomials, 2m and -7, it is a binomial.

Example 4 State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial. x 2 + 3x – The expression x 2 + 3x – 4 – 5 can be written as x 2 + 3x + (-9). So, it is a polynomial. Since it can be written as the sum of three monomials, it is a trinomial.

Example 5 State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial x The expression is not a polynomial since it is not a monomial. It contains division.

Your Turn State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial. 5a – Yes, Binomial

Your Turn State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial. 4m No, Cannot have a negative exponent

Your Turn State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial. 3y 2 – 6 + 7y Yes, Trinomial

The terms of a polynomial are usually arranged so that the powers of one variable are in descending or ascending order. PolynomialDescending OrderAscending Order 2x + x 2 + 1x 2 + 2x x + x 2 3y 2 + 5y 3 + y5y 3 + 3y 2 + yy + 3y 2 + 5y 3 x 2 + y 2 + 3xyx 2 + 3xy + y 2 y 2 + 3xy + x 2 2xy + y 2 + x 2 y 2 + 2xy + x 2 x 2 + 2xy + y 2

Degree The degree of a monomial is the sum of the exponents of the variables. MonomialDegree -3x 2 2 5pq = 3 20

To find the degree of a polynomial, you must find the degree of each term. The degree of the polynomial is the greatest of the degrees of its term. PolynomialTermsDegree of the Terms Degree of the Polynomial 2n + 72n, 71, 01 3x 2 + 5x3x 2, 5x2, 12 a 6 + 2a 3 + 1a 6, 2a 3, 16, 3, 06 5x 4 – 4a 2 b 6 + 3x5x 4, 4a 2 b 6, 3x4, 8, 18

Example 6 Find the degree of each polynomial. 5a So, the degree of 5a is 2. TermDegree 5a

Example 6 Find the degree of each polynomial. 6x 2 – 4x 2 y – 3xy So, the degree of 6x 2 – 4x 2 y – 3xy is 3. TermDegree 6x 2 2 4x 2 y2 + 1 or 3 3xy1 + 1 or 2

Your Turn Find the degree of each polynomial. 3x 2 – 7x 2

Your Turn Find the degree of each polynomial. 8m 3 – 2m 2 n