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Chapter 5 Section 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.4 1 2 3 4 5 6 Adding and Subtracting Polynomials; Graphing Simple Polynomials Identify terms and coefficients. Add like terms. Know the vocabulary for polynomials. Evaluate polynomials Add and subtract polynomials. Graph equations defined by polynomials of degree 2. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Identify terms and coefficients. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 3 Identify terms and coefficients. In Section 1.8, we saw that in an expression such as 4x3 6x2 5x 8, the quantities 4x3, 6x2, 5x, and 8 are called terms. In the term 4x3, the number 4 is called the numerical coefficient, or simply the coefficient, of x3. In the same way, 6 is the coefficient of x2 in the term 6x2, and 5 is the coefficient of x in the term 5x. The constant term 8 can be thought of as 8 · 1 = 8x2, since x0 = 1, so 8 is the coefficient in the term 8. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 4 EXAMPLE 1 Identifying Coefficients Name the coefficient of each term in the expression 2 x x. 3 Solution: 2, 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 5 Objective 2 Add like terms. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 6 Add like terms. Recall from Section 1.8 that like terms have exactly the same combinations of variables, with the same exponents on the variables. Only the coefficients may differ. 3 3 19m and 14m 9 9 9 6 y , 37 y , and y 3pq and 2 pq Examples of like terms 2xy 2 and xy 2 Using the distributive property, we combine, or add, like terms by adding their coefficients. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 7 EXAMPLE 2 Adding Like Terms Simplify by adding like terms. r 2 3r 5r 2 Solution: 6r 2 3r Unlike terms cannot be combined. Unlike terms have different variables or different exponents on the same variables. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 8 Objective 3 Know the vocabulary for polynomials. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 9 Know the vocabulary for polynomials. A polynomial in x is a term or the sum of a finite number of terms of the form axn, for any real number a and any whole number n. For example, 16 x8 7 x6 5x4 3x2 4 Polynomial is a polynomial in x. (The 4 can be written as 4x0.) This polynomial is written in descending powers of variable, since the exponents on x decrease from left to right. By contrast, 2x 3 x 2 1 x Not a Polynomial is not a polynomial in x, since a variable appears in a denominator. A polynomial could be defined using any variable and not just x. In fact, polynomials may have terms with more than one variable. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 10 Know the vocabulary for polynomials. (cont’d) The degree of a term is the sum of the exponents on the variables. For example 3x4 has degree 4, while the term 5x (or 5x1) has degree 1, −7 has degree 0 ( since −7 can be written −7x0), and 2x2y has degree 2 + 1 = 3. (y has an exponent of 1.) The degree of a polynomial is the greatest degree of any nonzero term of the polynomial. For example 3x4 + 5x2 + 6 is of degree 4, the term 3 (or 3x0) is of degree 0, and x2y + xy − 5xy2 is of degree 3. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 11 Know the vocabulary for polynomials. (cont’d) Three types of polynomials are common and given special names. A polynomial with only one term is called a monomial. (Mono means “one,” as in monorail.) Examples are 9m, 6y5 , a2 , and 6. monomials A polynomial with exactly two terms is called a binomial. (Bi- means “two,” as in bicycle.) Examples are 9 x4 9 x3 , 8m2 6m, and 3m5 9m2 . binomials A polynomial with exactly three terms is called a trinomial. (Tri- means “three,” as in triangle.) Examples are 19 2 8 y y 5, and 3m5 9m2 2. 9m 4m 6, 3 3 3 2 trinomials Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 12 EXAMPLE 3 Classifying Polynomials Simplify, give the degree, and tell whether the simplified polynomial is a monomial, binomial, trinomial, or none of these. x8 x 7 2 x8 Solution: 3x8 x7 degree 8; binomial Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 13 Objective 4 Evaluate polynomials. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 14 EXAMPLE 4 Evaluating a Polynomial Find the value of 2y3 + 8y − 6 when y = −1. Solution: 3 2 1 8 1 6 2 1 8 6 2 8 6 16 Use parentheses around the numbers that are being substituted for the variable, particularly when substituting a negative number for a variable that is raised to a power, or a sign error may result. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 15 Objective 5 Add and subtract polynomials. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 16 Add and subtract polynomials. Polynomials may be added, subtracted, multiplied, and divided. To add two polynomials, add like terms. In Section 1.5 the difference x − y as x + (−y). (We find the difference x − y by adding x and the opposite of y.) For example, 7 2 7 2 5 and 8 2 8 2 6. A similar method is used to subtract polynomials. To subtract two polynomials, change all the signs in the second polynomial and add the result to the first polynomial Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 17 EXAMPLE 5 Adding Polynomials Vertically Add. 4 x 3x 2 x and 6 x 2 x 3x x 2 2 x 5 and 4 x2 2 3 2 3 2 Solution: 4 x3 3x 2 2 x 3 2 + 6 x 2 x 3x 10x3 x2 x x2 2 x 5 2 2 + 4x 5x2 2 x 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 18 EXAMPLE 6 Add. Adding Polynomials Horizontally 4 2 4 2 2 x 6 x 7 3 x 5 x 2 Solution: x4 x2 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 19 EXAMPLE 7 Subtracting Polynomials Perform the subtractions. 2 2 7 y 11 y 8 3 y 4 y 6 2 2 7 y 11 y 8 3 y Solution: 4 y 6 10 y 15 y 2 2 3 2 3 2 2 y 7 y 4 y 6 14 y 6 y 2 y 5 . from 14 y 6 y 2 y 5 2 y 7 y 4 y 6 3 2 3 2 12 y3 y 2 6 y 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 20 EXAMPLE 8 Subtract. Subtracting Polynomials Vertically 14 y 6 y 2 y 6 2 y3 7 y 2 3 Solution: 2 14 y 6 y 2 y 3 2 6 + 2 y 7 y 3 2 12 y y 2 y 6 3 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 21 EXAMPLE 9 Subtracting Polynomials with More than One Variable Subtract. 3 2 2 3 2 2 5 m n 3 m n 4 mn 7 m n m n 6mn Solution: 5m3n 3m 2 n 2 4mn 7m3n m 2 n 2 6mn 2m3 n 4m 2 n 2 10mn Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 22 Objective 6 Graph equations defined by polynomials of degree 2. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 23 Graph equations defined by polynomials of degree 2. In Chapter 3, we introduced graphs of straight lines. These graphs were defined by linear equations (which are polynomial equations of degree 1). By plotting points selectively, we can graph polynomial equations of degree 2. The graph of y = x2 is the graph of a function, since each input x is related to just one output y. The curve in the figure below is called a parabola. The point (0,0), the lowest point on this graph, is called the vertex of the parabola. The vertical line through the vertex (the y-axis here) is called the axis of the parabola. The axis of a parabola is a line of symmetry for the graph. If the graph is folded on this line, the two halves will match. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 24 EXAMPLE 10 Graphing Equations Defined by Polynomials of Degree 2 Graph y = 2x2. Solution: All polynomials of degree 2 have parabolas as their graphs. When graphing, find points until the vertex and points on either side of it are located. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5.4 - 25