CHAPTER 12 Polynomials: Operations Slide 2Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 12.1Integers as Exponents 12.2Exponents and Scientific Notation 12.3Introduction to Polynomials 12.4Addition and Subtraction of Polynomials 12.5Multiplication of Polynomials 12.6Special Products 12.7Operations with Polynomials in Several Variables 12.8Division of Polynomials

OBJECTIVES 12.1 Integers as Exponents Slide 3Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. aTell the meaning of exponential notation. bEvaluate exponential expressions with exponents of 0 and 1. cEvaluate algebraic expressions containing exponents. dUse the product rule to multiply exponential expressions with like bases. eUse the quotient rule to divide exponential expressions with like bases.

OBJECTIVES 12.1 Integers as Exponents Slide 4Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. fExpress an exponential expression involving negative exponents with positive exponents.

12.1 Integers as Exponents a Tell the meaning of exponential notation. Slide 5Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. An expression for a power is called exponential notation.

EXAMPLE 12.1 Integers as Exponents a Tell the meaning of exponential notation. 1 Slide 6Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

12.1 Integers as Exponents a Tell the meaning of exponential notation. Slide 7Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. We read exponential notation as follows: a n is read the nth power of a, or simply a to the nth, or a to the n. We often read x 2 as “x-squared.”

12.1 Integers as Exponents EXPONENTS OF 0 AND 1 Slide 8Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

12.1 Integers as Exponents b Evaluate exponential expressions with exponents of 0 and 1. Slide 9Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. We consider 0 0 to be not defined.

EXAMPLE 12.1 Integers as Exponents b Evaluate exponential expressions with exponents of 0 and 1. 2 Slide 10Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 12.1 Integers as Exponents c Evaluate algebraic expressions containing exponents. 4Area of a Compact Disc Slide 11Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The standard compact disc used for software and music has a radius of 6 cm. Find the area of such a CD (ignoring the hole in the middle). “cm 2 ” means “square centimeters” and “ ” means “is approximately equal to.”

EXAMPLE 12.1 Integers as Exponents c Evaluate algebraic expressions containing exponents. 5 Slide 12Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. When we evaluate with a negative number, we often use extra parentheses to show the substitution.

12.1 Integers as Exponents THE PRODUCT RULE Slide 13Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. For any number a and any positive integers m and n, (When multiplying with exponential notation, if the bases are the same, keep the base and add the exponents.)

EXAMPLE 12.1 Integers as Exponents d Use the product rule to multiply exponential expressions with like bases. Slide 14Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

For any nonzero number a and any positive integers m and n, (When dividing with exponential notation, if the bases are the same, keep the base and subtract the exponent of the denominator from the exponent of the numerator.) 12.1 Integers as Exponents THE QUOTIENT RULE Slide 15Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 12.1 Integers as Exponents e Use the quotient rule to divide exponential expressions with like bases. 14 Slide 16Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Divide and simplify.

12.1 Integers as Exponents NEGATIVE EXPONENT Slide 17Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. For any real number a that is nonzero and any integer n,

EXAMPLE 12.1 Integers as Exponents f Express an exponential expression involving negative exponents with positive exponents. Slide 18Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Express using positive exponents. Then simplify.

EXAMPLE 12.1 Integers as Exponents f Express an exponential expression involving negative exponents with positive exponents. Slide 19Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Simplify. Write the result using positive exponents.

12.1 Integers as Exponents Slide 20Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.