Slide 1

Report
Chapter 5
Section 4
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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.
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Objective 1
Identify terms and coefficients.
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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.
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Slide 5.4 - 4
EXAMPLE 1
Identifying Coefficients
Name the coefficient of each term in the expression
2 x  x.
3
Solution:
2, 1
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Slide 5.4 - 5
Objective 2
Add like terms.
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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.
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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.
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Slide 5.4 - 8
Objective 3
Know the vocabulary for
polynomials.
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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.
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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.
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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
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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
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Slide 5.4 - 13
Objective 4
Evaluate polynomials.
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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.
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Slide 5.4 - 15
Objective 5
Add and subtract polynomials.
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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
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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
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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
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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
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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
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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 
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Objective 6
Graph equations defined by
polynomials of degree 2.
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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.
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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.
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Slide 5.4 - 25

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