Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 12 Exponents and Polynomials

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

Martin-Gay, Developmental Mathematics, 2e 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Defining Term and Coefficient Term – a number or a product of a number and variables raised to powers Coefficient – numerical factor of a term Constant – term which is only a number A Polynomial in x is a finite sum of terms of the form ax n, where a is a real number and n is a whole number.

Martin-Gay, Developmental Mathematics, 2e 44 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. Defining Term and Coefficient

Martin-Gay, Developmental Mathematics, 2e 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Monomial is a polynomial with exactly one term. Binomial is a polynomial with exactly two terms. Trinomial is a polynomial with exactly three terms. Types of Polynomials

Martin-Gay, Developmental Mathematics, 2e 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The degree of a term in one variable is the exponent on the variable. The degree of a constant is 0. The degree of a polynomial is the greatest degree of any term of the polynomial. The degree of 9x 3 – 4x is 3. The degree of a term with more than one variable is the sum of the exponents on the variables. The degree of the term 5a 4 b 3 c is 8 (remember that c can be written as c 1 ). Degrees

Martin-Gay, Developmental Mathematics, 2e 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Evaluating a polynomial for a particular value involves replacing the value for the variable(s) involved. Find the value of 2x 3 – 3x + 4 when x = –2. = 2( – 2) 3 – 3( – 2) + 42x 3 – 3x + 4 = 2( – 8) = – 6 Evaluating Polynomials Example:

Martin-Gay, Developmental Mathematics, 2e 88 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. Combine like terms to simplify. x 2 y + xy – y + 10x 2 y – 2y + xy Only like terms can be combined through addition and subtraction. Warning! = 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 Group like terms. Combining Like Terms Example Use the distributive property. Simplify.