CHAPTER 4 Polynomials: Operations Slide 2Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 4.1Integers as Exponents 4.2Exponents and Scientific Notation 4.3Introduction to Polynomials 4.4Addition and Subtraction of Polynomials 4.5Multiplication of Polynomials 4.6Special Products 4.7Operations with Polynomials in Several Variables 4.8Division of Polynomials

OBJECTIVES 4.3 Introduction to Polynomials (continued) Slide 3Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. aEvaluate a polynomial for a given value of the variable. bIdentify the terms of a polynomial. cIdentify the like terms of a polynomial. dIdentify the coefficients of a polynomial. eCollect the like terms of a polynomial. fArrange a polynomial in descending order, or collect the like terms and then arrange in descending order.

OBJECTIVES 4.3 Introduction to Polynomials Slide 4Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. gIdentify the degree of each term of a polynomial and the degree of the polynomial. hIdentify the missing terms of a polynomial. iClassify a polynomial as a monomial, a binomial, a trinomial, or none of these.

Examples: 3x 2 2 2x 3x 6 0 A monomial is an expression of the type axn, where a is a real number constant and n is a nonnegative integer. 4.3 Introduction to Polynomials Monomial Slide 5Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

Examples: 5w + 8,  3x 2 + x + 4, x, 0, 75y 6 A polynomial is a monomial or a combination of sums and/or differences of monomials. 4.3 Introduction to Polynomials Polynomial Slide 6Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

When we replace the variable in a polynomial with a number, the polynomial then represents a number called a value of the polynomial. Finding that number, or value, is called evaluating the polynomial. 4.3 Introduction to Polynomials a Evaluate a polynomial for a given value of the variable. Slide 7Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE a. 5x + 2b. 3x 2 – 4x + 1 Solution a. 5x + 2 = 5 · = = 17 b. 3x 2 – 4x + 1 = 3 · 3 2 – 4 · = 3 · 9 – 4 · = 27 – = Introduction to Polynomials a Evaluate a polynomial for a given value of the variable. AEvaluate the polynomial when x = 3. Slide 8Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Solution For x =  3, we have  x 3 + 4x + 7 =  (  3) 3 + 4(  3) + 7 =  (  27) + 4(  3) + 7 = 27 + (  12) + 7 = Introduction to Polynomials a Evaluate a polynomial for a given value of the variable. B Evaluate  x 3 + 4x + 7 for x =  3. Slide 9Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE In a sports league of n teams in which each team plays every other team twice, the total number of games to be played is given by the polynomial n 2  n. A boys’ soccer league has 12 teams. How many games are played if each team plays every other team twice? 4.3 Introduction to Polynomials a Evaluate a polynomial for a given value of the variable. CApplications of Polynomials (continued) Slide 10Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Solution: We evaluate the polynomial for n = 12: n 2  n = 12 2  12 = 144  12 = 132. The league plays 132 games. 4.3 Introduction to Polynomials a Evaluate a polynomial for a given value of the variable. CApplications of Polynomials Slide 11Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Solution The terms are written as 7p 5,  3p 3, and 3. 7p 5  3p These are the terms of the polynomial. 4.3 Introduction to Polynomials b Identify the terms of a polynomial. DIdentify the terms of the polynomial 7p 5  3p Slide 12Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

When terms have the same variable and the variable is raised to the same power, we say that they are like terms. 4.3 Introduction to Polynomials c Identify the like terms of a polynomial. Slide 13Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE a. 5x 3 + 6x – 3x 2 + 2x 3 + x 2 b. 8 – 7a 2 – 9 – a – 2a Solution a. 5x 3 + 6x – 3x 2 + 2x 3 + x 2 Like terms: 2x 3 and 5x 3 Same variable and exponent Like terms: –3x 2 and x 2 Same variable and exponent b. 8 – 7a 2 – 9 – a – 2a Like terms: –a and –2a, 8 and –9 4.3 Introduction to Polynomials c Identify the like terms of a polynomial. EIdentify all the like terms in the polynomials. Slide 14Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

The part of a term that is a constant factor is the coefficient of that term. The coefficient of 4y is Introduction to Polynomials d Identify the coefficients of a polynomial. Slide 15Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE 5x4 – 8x 2 y + y – 9 Solution The coefficient of 5x 4 is 5. The coefficient of –8x 2 y is –8. The coefficient of y is 1, since y = 1y. The coefficient of –9 is simply – Introduction to Polynomials d Identify the coefficients of a polynomial. FIdentify the coefficient of each term in the polynomial. Slide 16Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE a) 4y 4  9y 4 b) 7x x 2 + 6x 2  13  6x 5 c) 9w 5  7w w 5 + 2w 3 Solution a) 4y 4  9y 4 = (4  9)y 4 =  5y 4 b) 7x x 2 + 6x 2  13  6x 5 = 7x 5  6x 5 + 3x 2 + 6x  13 = x 5 + 9x 2  Introduction to Polynomials e Collect the like terms of a polynomial. GCombine like terms. (continued) Slide 17Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE a) 4y 4  9y 4 b) 7x x 2 + 6x 2  13  6x 5 c) 9w 5  7w w 5 + 2w 3 Solution c) 9w 5  7w w 5 + 2w 3 = 9w w 5  7w 3 + 2w 3 = 20w 5  5w Introduction to Polynomials e Collect the like terms of a polynomial. GCombine like terms. Slide 18Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE We usually arrange polynomials in descending order by exponent, but not always. The opposite order is called ascending order. 7x 5 + 5x 7 + x 2 + 3x 3 Solution: 7x 5 + 5x 7 + x 2 + 3x 3 = 5x 7 + 7x 5 + 3x 3 + x Introduction to Polynomials f Arrange a polynomial in descending order, or collect the like terms and then arrange in descending order. FArrange the polynomial in descending order. Slide 19Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

The degree of a term of a polynomial is the number of variable factors in that term. 4.3 Introduction to Polynomials g Identify the degree of each term of a polynomial and the degree of the polynomial. Slide 20Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE a) 9x 5 b) 6y c) 9 Solution a) The degree of 9x 5 is 5. b) The degree of 6y is 1. c) The degree of 9 is Introduction to Polynomials g Identify the degree of each term of a polynomial and the degree of the polynomial. IDetermine the degree of each term: Slide 21Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

The degree of the polynomial is the largest of the degrees of the terms, unless it is a polynomial Introduction to Polynomials g Identify the degree of each term of a polynomial and the degree of the polynomial. Slide 22Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE 4x 2  9x 3 + 6x 4 + 8x  7. Solution The largest exponent is 4. The degree of the polynomial is Introduction to Polynomials g Identify the degree of each term of a polynomial and the degree of the polynomial. JIdentify the degree of the polynomial. Slide 23Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

If a coefficient is 0, we generally do no write the term. We say that we have a missing term. 4.3 Introduction to Polynomials h Identify the missing terms of a polynomial. Slide 24Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE 8x 6 – 3x 3 + 5x 2 + 9x – 8 Solution There is no x 5 or x 4 term so those two terms are missing. 8x 6 + 0x 5 + 0x 4 – 3x 3 + 5x 2 + 9x – Introduction to Polynomials h Identify the missing terms of a polynomial. KIdentify the missing terms in the polynomial. Slide 25Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Solution: x 4 – 3x 3 + 5x – 8 Missing terms:x 4 – 3x 3 + 0x 2 + 5x – 8 Leaving space:x 4 – 3x 3 + 5x – Introduction to Polynomials h Identify the missing terms of a polynomial. LWrite the polynomial x 4 – 3x 3 + 5x – 8 in two ways: with its missing terms and by leaving space for them. Slide 26Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

MonomialsBinomialsTrinomialsNone of These 5x25x2 3x + 43x 2 + 5x + 9 5x 3  6x 2 + 2xy  9 84a 5 + 7bc 7x 7  9z 3 + 5a 4 + 2a 3  a 2 + 7a  2  8a 23 b 3  10x 3  76x 2  4x  ½6x 6  4x 5 + 2x 4  x 3 + 3x  2 A polynomial that is composed of two terms is called a binomial, whereas those composed of three terms are called trinomials. Polynomials with four or more terms have no special name. 4.3 Introduction to Polynomials i Classify a polynomial as a monomial, a binomial, a trinomial, or none of these. Slide 27Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.