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Chapter 5 Section 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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Adding and Subtracting Polynomials; Graphing Simple Polynomials 1 1 4 4 3 3 2 2 5.45.4 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. 6 6 5 5

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Slide 5.4 - 3 Identify terms and coefficients.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Identify terms and coefficients. Slide 5.4 - 4 In Section 1.8, we saw that in an expression such as the quantities 4x 3, 6x 2, 5x, and 8 are called terms. In the term 4x 3, the number 4 is called the numerical coefficient, or simply the coefficient, of x 3. In the same way, 6 is the coefficient of x 2 in the term 6x 2, and 5 is the coefficient of x in the term 5x. The constant term 8 can be thought of as 8 · 1 = 8x 2, since x 0 = 1, so 8 is the coefficient in the term 8.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Name the coefficient of each term in the expression EXAMPLE 1 Identifying Coefficients Solution: Slide 5.4 - 5

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Add like terms. Slide 5.4 - 6

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. Using the distributive property, we combine, or add, like terms by adding their coefficients. Slide 5.4 - 7 Examples of like terms

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Solution: Adding Like Terms Slide 5.4 - 8 Simplify by adding like terms. Unlike terms cannot be combined. Unlike terms have different variables or different exponents on the same variables.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 Objective 3 Know the vocabulary for polynomials. Slide 5.4 - 9

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Know the vocabulary for polynomials. A polynomial in x is a term or the sum of a finite number of terms of the form ax n, for any real number a and any whole number n. For example, is a polynomial in x. (The 4 can be written as 4x 0.) This polynomial is written in descending powers of variable, since the exponents on x decrease from left to right. By contrast, 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. Slide 5.4 - 10 Polynomial Not a Polynomial

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Know the vocabulary for polynomials. (cont’d) The degree of a term is the sum of the exponents on the variables. For example 3x 4 has degree 4, while the term 5x (or 5x 1 ) has degree 1, −7 has degree 0 ( since −7 can be written −7x 0 ), and 2x 2 y has degree 2 + 1 = 3. (y has an exponent of 1.) Slide 5.4 - 11 The degree of a polynomial is the greatest degree of any nonzero term of the polynomial. For example 3x 4 + 5x 2 + 6 is of degree 4, the term 3 (or 3x 0 ) is of degree 0, and x 2 y + xy − 5xy 2 is of degree 3.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Know the vocabulary for polynomials. (cont’d) Slide 5.4 - 12 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 and monomials A polynomial with exactly two terms is called a binomial. (Bi- means “two,” as in bicycle.) Examples are and binomials A polynomial with exactly three terms is called a trinomial. (Tri- means “three,” as in triangle.) Examples are and trinomials

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solution: Classifying Polynomials Slide 5.4 - 13 Simplify, give the degree, and tell whether the simplified polynomial is a monomial, binomial, trinomial, or none of these. degree 8; binomial

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4 Objective 4 Evaluate polynomials. Slide 5.4 - 14

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solution: Evaluating a Polynomial Slide 5.4 - 15 Find the value of 2y 3 + 8y − 6 when y = −1. 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.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5 Objective 5 Add and subtract polynomials. Slide 5.4 - 16

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Add and subtract polynomials. Slide 5.4 - 17 Polynomials may be added, subtracted, multiplied, and divided. To subtract two polynomials, change all the signs in the second polynomial and add the result to the first polynomial 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, and A similar method is used to subtract polynomials.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Add. and EXAMPLE 5 Solution: Adding Polynomials Vertically Slide 5.4 - 18 + +

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Solution: Adding Polynomials Horizontally Slide 5.4 - 19 Add.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Perform the subtractions. from EXAMPLE 7 Solution: Subtracting Polynomials Slide 5.4 - 20

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Subtract. EXAMPLE 8 Solution: Subtracting Polynomials Vertically Slide 5.4 - 21 +

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Subtracting Polynomials with More than One Variable Slide 5.4 - 22 Subtract. Solution:

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6 Objective 6 Graph equations defined by polynomials of degree 2. Slide 5.4 - 23

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graph equations defined by polynomials of degree 2. Slide 5.4 - 24 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 = x 2 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.

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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 10 Graphing Equations Defined by Polynomials of Degree 2 Slide 5.4 - 25 Graph y = 2x 2. 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.

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