Slide 7- 1 Copyright © 2012 Pearson Education, Inc.

5.1 Introduction to Polynomials and Polynomial Functions ■ Terms and Polynomials ■ Degree and Coefficients ■ Polynomial Functions ■ Graphs of Polynomial Functions ■ Adding Polynomials ■ Opposites and Subtraction

Slide 5- 3 Copyright © 2012 Pearson Education, Inc. Characteristics of Polynomial Functions The graph of a polynomial function is “smooth”, that is, there are no sharp corners. The graph of a polynomial function is continuous, that is, there are no holes or breaks. The domain of a polynomial function, unless otherwise specified, is all real numbers. The domain of a polynomial function is (–∞, ∞).

Slide 5- 4 Copyright © 2012 Pearson Education, Inc. Algebraic Expressions and Polynomials A term can be a number, a variable, a product of numbers and/or variables, or a quotient of numbers and/or variables. A term that is a product of constants and/or variables is called a monomial. Examples of monomials: 8, w, 24 x 3 y A polynomial is a monomial or a sum of monomials. Examples of polynomials: 5w + 8,  3x 2 + x + 4, x, 0, 75y 6

Slide 5- 5 Copyright © 2012 Pearson Education, Inc. The leading term of a polynomial is the term of highest degree. Its coefficient is called the leading coefficient and its degree is referred to as the degree of the polynomial. Consider this polynomial 4x 2  9x 3 + 6x 4 + 8x  7. The terms are 4x 2,  9x 3, 6x 4, 8x, and  7. The coefficients are 4,  9, 6, 8 and  7. The degree of each term is 2, 3, 4, 1, and 0. The leading term is 6x 4 and the leading coefficient is 6. The degree of the polynomial is 4.

Slide 5- 6 Copyright © 2012 Pearson Education, Inc. 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. MonomialsBinomialsTrinomialsPolynomials 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

Slide 5- 7 Copyright © 2012 Pearson Education, Inc. Polynomial Functions If the polynomial describing a function has degree 1, we say that the function is linear. A quadratic function is described by a polynomial of degree 2, a cubic function by a polynomial of degree 3,and a quartic function by a polynomial of degree 4.

Slide 5- 8 Copyright © 2012 Pearson Education, Inc. Example Find P(  3) for the polynomial function given by P(x) =  x 3 + 4x + 7. Solution P(  3) =  (  3) 3 + 4(  3) + 7 =  (  27) + 4(  3) + 7 = 27 + (  12) + 7 = 22

Slide 5- 9 Copyright © 2012 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? Solution We evaluate the polynomial for n = 12: n 2  n = 12 2  12 = 144  12 = 132. The league plays 132 games.

Slide Copyright © 2012 Pearson Education, Inc. Example The average number of accidents per day involving drivers of age r can be approximated by the polynomial f(r) = 0.4r 2  40r Find the average number of accidents per day involving 25-year- old drivers. Solution f(25) = 0.4(25) 2  40(25) = 0.4(625)  = 250  = 289 There are, on average, approximately 289 accidents each day involving 25-year-old drivers.

Slide Copyright © 2012 Pearson Education, Inc. Graph the polynomial function in the standard view and estimate the range of the graph. f(x) = x 3 – 3x Solution Example The range is all real numbers.

Slide Copyright © 2012 Pearson Education, Inc. Adding Polynomials

Slide Copyright © 2012 Pearson Education, Inc. Example Combine like terms. 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  4 c) 9w 5  7w w 5 + 2w 3 = 9w w 5  7w 3 + 2w 3 = 20w 5  5w 3 Adding Polynomials

Slide Copyright © 2012 Pearson Education, Inc. Example Add: (  6x 3 + 7x  2) + (5x 3 + 4x 2 + 3). Solution (  6x 3 + 7x  2) + (5x 3 + 4x 2 + 3) = (  6 + 5)x 3 + 4x 2 + 7x + (  2 + 3) =  x 3 + 4x 2 + 7x + 1

Slide Copyright © 2012 Pearson Education, Inc. Add: (3  4x + 2x 2 ) + (  6 + 8x  4x 2 + 2x 3 ). Solution (3  4x + 2x 2 ) + (  6 + 8x  4x 2 + 2x 3 ) = (3  6) + (  4 + 8)x + (2  4)x 2 + 2x 3 =  3 + 4x  2x 2 + 2x 3 Example

Slide Copyright © 2012 Pearson Education, Inc. Add: 10x 5  3x 3 + 7x and 6x 4  8x and 4x 6  6x 5 + 2x Solution 10x 5  3x 3 + 7x x 4  8x x 6  6x 5 + 2x x 6 + 4x 5 + 6x 4  3x 3 + x The answer is 4x 6 + 4x 5 + 6x 4  3x 3 + x Example

Slide Copyright © 2012 Pearson Education, Inc. The Opposite of a Polynomial The opposite of a polynomial P can be written as ‒ P or, equivalently, by replacing each terms with its opposite.

Slide Copyright © 2012 Pearson Education, Inc. Simplify:  (  8x 4  x 3 + 9x 2  2x + 72). Solution  (  8x 4  x 3 + 9x 2  2x + 72) = 8x 4 + x 3  9x 2 + 2x  72 Example

Slide Copyright © 2012 Pearson Education, Inc. (10x 5 + 2x 3  3x 2 + 5)  (  3x 5 + 2x 4  5x 3  4x 2 ). Solution (10x 5 + 2x 3  3x 2 + 5)  (  3x 5 + 2x 4  5x 3  4x 2 ) = 10x 5 + 2x 3  3x x 5  2x 4 + 5x 3 + 4x 2 = 13x 5  2x 4 + 7x 3 + x Example Subtract:

Slide Copyright © 2012 Pearson Education, Inc. Subtract: (8x 5 + 2x 3  10x)  (4x 5  5x 3 + 6). Solution (8x 5 + 2x 3  10x)  (4x 5  5x 3 + 6) = 8x 5 + 2x 3  10x + (  4x 5 ) + 5x 3  6 = 4x 5 + 7x 3  10x  6 Example

Slide Copyright © 2012 Pearson Education, Inc. Write in columns and subtract: (6x 2  4x + 7)  (10x 2  6x  4). Solution 6x 2  4x + 7  (10x 2  6x  4)  4x 2 + 2x + 11 Example