Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

Chapter 6 Exponents, Polynomials, and Polynomial Functions

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 6.3 Polynomials and Polynomial Functions

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Polynomial Vocabulary Term – a number or the product of a number and one or more variables raised to powers Coefficient – numerical factor of a term Constant – term which is only a number Polynomial is a sum of terms involving variables raised to a whole number exponent, with no variables appearing in any denominator.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall In the polynomial 7x 5 + x 2 y 2 – 4xy + 7 There are 4 terms: 7x 5, x 2 y 2, –4xy and 7. The coefficient of term 7x 5 is 7, of term x 2 y 2 is 1, of term –4xy is –4 and of term 7 is 7. 7 is a constant term. Polynomial Vocabulary

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Monomial is a polynomial with one term. Binomial is a polynomial with two terms. Trinomial is a polynomial with three terms. Types of Polynomials

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Degree of a Term The degree of a term is the sum of the exponents on the variables contained in the term. Degree of a Polynomial The degree of a polynomial is the largest degree of all its terms.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Find the degree of each term. a. 5x 7 Example b. −4 3 x 6 c Solution a. 5x 7 The exponent on x is 7, so the degree of the term is 7. b. −4 3 x 6 The exponent on x is 6, so the degree of the term is 6. c. The degree of 7.26, which can be written as 7.26x 0, is 0.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Find the degree of each polynomial and indicate whether the polynomial is also a monomial, binomial, or trinomial. Example PolynomialDegreeClassification 3Trinomial = 4Monomial 6Binomial

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Find the degree of the polynomial Example Degree: The largest degree of any term is 5, so the degree of this polynomial is 5. Solution

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall If P(x) = 4x 2 − 3x + 1. Find the following. a. P(2) Example b. P(−3) Solution a. Substitute 2 in for x and simplify. P(x) = 4x 2 − 3x + 1 P(2) = 4(2) 2 − 3(2) + 1 = 11 b. Substitute −3 in for x and simplify. P(−3) = 4(−3) 2 − 3(−3) + 1 = 46

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Like terms are terms that contain exactly the same variables raised to exactly the same powers. Only like terms can be combined through addition and subtraction. Warning! Like TermsUnlike Terms −7x 2, −x 2 5x 2, −x 8x 2 y 4 z, 5x 2 zy 4 6x 2 y 3, −4xy 2

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Combine like terms to simplify. x 2 y + xy – y + 10x 2 y – 2y + xy = 11x 2 y + 2xy – 3y = (1 + 10)x 2 y + (1 + 1)xy + (–1 – 2)y = x 2 y + 10x 2 y + xy + xy – y – 2y Example Solution

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Adding Polynomials To add polynomials, combine all like terms. Subtracting Polynomials To subtract a polynomial, add its opposite.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Add. (3x – 8) + (4x 2 – 3x +3) = 4x 2 + 3x – 3x – = 4x 2 – 5 = 3x – 8 + 4x 2 – 3x + 3 Example Solution Remove parentheses and group like terms. (3x – 8) + (4x 2 – 3x +3)

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Subtract. 4 – (– y – 4) = 4 + y + 4 = y = y + 8 Example Solution To subtract, add the opposite of the second polynomial to the first. 4 – (– y – 4)

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Simplify. = 3a 2 – 6a + 11 (– a 2 + 1) – (a 2 – 3) + (5a 2 – 6a + 7) = –a – a a 2 – 6a + 7 = –a 2 – a 2 + 5a 2 – 6a Example Solution (– a 2 + 1) – (a 2 – 3) + (5a 2 – 6a + 7)

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall In the previous examples, after discarding the parentheses, we would rearrange the terms so that like terms were next to each other in the expression. You can also use a vertical format in arranging your problem, so that like terms are aligned with each other vertically.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Using the degree of a polynomial, we can determine what the general shape of the function will be, before we ever graph the function. A polynomial function of degree 1 is a linear function. We have examined the graphs of linear functions in great detail previously in this course and prior courses. A polynomial function of degree 2 is a quadratic function. In general, for the quadratic equation of the form y = ax 2 + bx + c, the graph is a parabola opening up when a > 0, and opening down when a < 0. a > 0 x a < 0 x Types of Polynomials

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Polynomial functions of degree 3 are cubic functions. Cubic functions have four different forms, depending on the coefficient of the x 3 term. x 3 coefficient is positive x 3 coefficient is negative