4.4 Adding and Subtracting Polynomials; Graphing Simple Polynomials

Objective 1

Identify terms and coefficients.

Slide 4.4-3

Identify terms and coefficients.

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

the quantities 4x3, 6x2, 5x, and 8 are called terms. In the first (or leading) 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

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Solution:

2, 1

Name the coefficient of each term in the expression32 .x x

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Identifying CoefficientsCLASSROOM EXAMPLE 1

Objective 2

Add like terms.

Slide 4.4-6

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.

3 3and 19 14m m

Examples of like terms

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

2 2and 2xy xy

Slide 4.4-7

Solution:

2 23 5r r r

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.

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Adding Like TermsCLASSROOM EXAMPLE 2

Objective 3

Know the vocabulary for polynomials.

Slide 4.4-9

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.

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

3 2 12x x

x

Polynomial in x

Not a Polynomial

Slide 4.4-10

A polynomial can be defined using any variable and not just x. In fact, polynomials may have terms with more than one variable.

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.

Slide 4.4-11

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

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 mand

Binomials

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

TrinomialsSlide 4.4-12

Know the vocabulary for polynomials. (cont’d)

Solution:

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

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Classifying PolynomialsCLASSROOM EXAMPLE 3

Objective 4

Evaluate polynomials.

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Solution:

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.

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Evaluating a PolynomialCLASSROOM EXAMPLE 4

Objective 5

Add and subtract polynomials.

Slide 4.4-16

Add and subtract polynomials.

Polynomials may be added, subtracted, multiplied, and divided.

Subtracting PolynomialsTo subtract two polynomials, change all the signs in the second polynomial and add the result to the first polynomial.

Adding PolynomialsTo add two polynomials, add like terms.

In Section 1.5, we defined 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

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Add.

and

and

Solution:

3 24 3 2x x x 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

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Adding Polynomials VerticallyCLASSROOM EXAMPLE 5

Solution:

Add.

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

4 2 9x x

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Adding Polynomials HorizontallyCLASSROOM EXAMPLE 6

Perform the subtractions.

from

Solution:

3 214 6 2 5 .y y y

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

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Subtracting Polynomials HorizontallyCLASSROOM EXAMPLE 7

Subtract.

Solution:

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

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Subtracting Polynomials VerticallyCLASSROOM EXAMPLE 8

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

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Adding and Subtracting Polynomials with More Than One VariableCLASSROOM EXAMPLE 9

Objective 6

Graph equations defined by polynomials of degree 2.

Slide 4.4-23

Graph equations defined by polynomials of degree 2.

In Chapter 3, we introduced graphs 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 to the right 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.

Slide 4.4-24

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.

Slide 4.4-25

Graphing Equations Defined by Polynomials of Degree 2CLASSROOM EXAMPLE 10