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

Chapter Chapter 55Section Section 44

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


Adding and Subtracting Polynomials; Graphing Simple Polynomials

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5.45.45.45.4Identify terms and coefficients.Add like terms.Know the vocabulary for polynomials.Evaluate polynomialsAdd and subtract polynomials.Graph equations defined by polynomials of degree 2.

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Objective 11

Slide 5.4 - 3

Identify terms and coefficients.


Identify terms and coefficients.

Slide 5.4 - 4

In Section 1.8, we saw that in an expression such as

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 8.3 24 ,6 5 8x x x


Name the coefficient of each term in the expression

EXAMPLE 1 Identifying Coefficients

Solution:

Slide 5.4 - 5

2, 1

32 .x x


Objective 22

Add like terms.

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.

Using the distributive property, we combine, or add, like terms by adding their coefficients.

Slide 5.4 - 7

3 3and 19 14m m

Examples of like terms

9 9 9, , and 6 37y y yand 2 3pq pq

2 2and 2xy xy


EXAMPLE 2

Solution:

Adding Like Terms

2 23 5r r r

Slide 5.4 - 8

Simplify by adding like terms.

26 3r r

Unlike terms cannot be combined. Unlike terms have different variables or different exponents on the same variables.


Objective 33

Know the vocabulary for polynomials.

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,

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,

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

8 6 4 216 7 5 3 4x x x x

3 2 12x x

x

Polynomial

Not a Polynomial


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.)

Slide 5.4 - 11

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.


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

6.9 ,m 5 ,6y 2 ,a andmonomials

A polynomial with exactly two terms is called a binomial. (Bi- means “two,” as in bicycle.) Examples are

4 39 9 ,x x 28 6 ,m m 5 23 .9m mandbinomials

A polynomial with exactly three terms is called a trinomial. (Tri- means “three,” as in triangle.) Examples are

3 2 6,9 4m m 219 8,5

3 3y y 5 23 9 .2m m and

trinomials


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.

8 7 82x x x

8 73x x

degree 8; binomial


Objective 44

Evaluate polynomials.

Slide 5.4 - 14


EXAMPLE 4

Solution:

Evaluating a Polynomial

Slide 5.4 - 15

Find the value of 2y3 + 8y − 6 when y = −1.

312 61 8

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.


Objective 55

Add and subtract polynomials.

Slide 5.4 - 16


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.

27 2 7 5 8 8 6.2 2


Add.

and

and

EXAMPLE 5

Solution:

Adding Polynomials Vertically

3 24 3 2x x x

Slide 5.4 - 18

2 2 5x x

3 26 2 3x x x 24 2x

3 24 3 2x x x 3 26 2 3x x x +

2 2 5x x 24x 2+

3 210x x x 25 2 3x x


EXAMPLE 6

Solution:

Adding Polynomials Horizontally

Slide 5.4 - 19

Add.

4 2 4 22 6 7 3 5 2x x x x

4 2 9x x


Perform the subtractions.

from

EXAMPLE 7

Solution:

Subtracting Polynomials

3 214 6 2 5 .y y y

Slide 5.4 - 20

2 27 11 8 3 4 6y y y y

2 27 11 8 3 4 6y y y y 210 15 2y y

3 2 3 214 6 2 5 2 7 4 6y y y y y y 3 212 6 11y y y

3 22 7 4 6y y y


Subtract.

EXAMPLE 8

Solution:

Subtracting Polynomials Vertically

Slide 5.4 - 21

3 214 6 2y y y 3 22 7y y 6

3 214 6 2y y y 3 22 7y y + 63 212 2 6y y y


EXAMPLE 9 Subtracting Polynomials with More than One Variable

Slide 5.4 - 22

Subtract.

3 2 2 3 2 25 3 4 7 6m n m n mn m n m n mn

Solution:

3 2 2 3 2 25 3 4 7 6m n m n mn m n m n mn

3 2 22 4 10m n m n mn


Objective 66

Graph equations defined by polynomials of degree 2.

Slide 5.4 - 23


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 = 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.


EXAMPLE 10 Graphing Equations Defined by Polynomials of Degree 2

Slide 5.4 - 25

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.

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

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