Transcript Chapter 5 Polynomials, Polynomial Functions, and Factoring § 5.1 Introduction to Polynomials and Polynomial Functions Polynomials A polynomial is a single term or the sum of two.

Chapter 5
Polynomials, Polynomial
Functions, and Factoring
§ 5.1
Introduction to Polynomials and Polynomial
Functions
Polynomials
A polynomial is a single term or the sum of two or more
terms containing variables with whole number exponents.
Consider the polynomial:
3x  2 x  5 x  6
4
3
This polynomial contains four terms. It is customary to write the terms in
order of descending powers of the variable. This is the standard form of a
polynomial.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 5.1
Polynomials
The degree of a polynomial is the greatest degree of any term
of the polynomial. The degree of a term axn y m is (n +m)
and the coefficient of the term is a. If there is exactly one
term of greatest degree, it is called the leading term. It’ s
coefficient is called the leading coefficient. Consider the
polynomial: 3x 4  2 x 3  5 x  6
3 is the leading coefficient. The degree is 4.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 5.1
Polynomials
The Degree of ax n T
If a  0, the degree of ax is n. The degree of a
nonzero constant is 0. The constant 0 has no defined
degree.
n
Adding Polynomials
Polynomials are added by removing the parentheses that
surround each polynomial (if any) and then combining
like terms.
Subtracting Polynomials
To subtract two polynomials, change the sign of every
term of the second polynomial. Add this result to the
first polynomial.
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 5.1
Polynomials
EXAMPLE
Determine the coefficient of each term, the degree of each
term, the degree of the polynomial, the leading term, and the
leading coefficient of the polynomial.
12x 4 y  5x3 y 7  x 2  4
SOLUTION
Term Coefficient
Degree (Sum of Exponents on
the Variables)
12x 4 y
 5x3 y 7
 x2
12
4+1=5
-5
3 + 7 = 10
-1
2+0=2
4
4
0+0=0
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 5.1
Polynomials
CONTINUED
The degree of the polynomial is the greatest degree of all its
terms, which is 10. The leading term is the term of the
greatest degree, which is  5x3 y 7. Its coefficient, -5, is the
leading coefficient.
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 5.1
Polynomials
f ( x)  4x  2x  5x  2
3
2
is an example of a polynomial function. In a polynomial
function, the expression that defines the function is a
polynomial.
How do you evaluate a polynomial function? Use
substitution just as you did to evaluate functions in Chapter 2.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 5.1
Polynomials
EXAMPLE
The polynomial function
f x  2212x 2  57,575x  107,896
models the cumulative number of deaths from AIDS in the
United States, f (x), x years after 1990. Use this function to
solve the following problem.
Find and interpret f (8).
SOLUTION
To find f (8), we replace each occurrence of x in the function’s
formula with 8.
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 5.1
Polynomials
CONTINUED
f x  2212x 2  57,575x  107,896
f 8  141,568 460,600 107,896
Original function
Replace each occurrence
of x with 8
Evaluate exponents
Multiply
f 8  426,928
Add
f 8  22128  57,5758 107,896
2
f 8  221264  57,5758  107,896
Thus, f (8) = 426,928. According to this model, this means that
8 years after 1990, in 1998, there had been 426,928 cumulative
deaths from AIDS in the United States.
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 5.1
Polynomials
Polynomial functions of degree 2 or higher have graphs
that are smooth and continuous.
By smooth, we mean that the graph contains only
rounded corners with no sharp corners.
By continuous, we mean that the graph has no breaks
and can be drawn without lifting the pencil from the
rectangular coordinate system.
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 5.1
Graphs of Polynomials
EXAMPLE
Smooth rounded
curve
Sharp Corner

Discontinuous
break
The graph below does not represent
a polynomial function. Although it
has a couple of smooth, rounded
corners, it also has a sharp corner
and a break in the graph. Either one
of these last two features disqualifies
it from being a polynomial function.
Smooth rounded
curve
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 5.1
Graphs of Polynomials
The Leading Coefficient Test
As x increases or decreases without bound, the graph of a polynomial function
eventually rises or falls. In particular,
Odd-Degree
Polynomials
Even-Degree
Polynomials
If the leading coefficient is positive, the graph falls to
the left and rises to the right.
If the leading coefficient is negative, the graph rises to
the left and falls to the right.
If the leading coefficient is positive, the graph rises to
the left and to the right.
If the leading coefficient is negative, the graph falls to
the left and to the right.
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 5.1
Polynomials
EXAMPLE
The common cold is caused by a rhinovirus. After x days of
invasion by the viral particles, the number of particles in our
bodies, f (x), in billions, can be modeled by the polynomial
function f x  0.75x4  3x3  5.
Use the Leading Coefficient Test to determine the graph’s end
behavior to the right. What does this mean about the number
of viral particles in our bodies over time?
SOLUTION
Since the polynomial function has even degree and has a
negative leading coefficient, the graph falls to the right (and
the left). This means that the viral particles eventually
decrease as the days increase.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 5.1
Adding Polynomials
EXAMPLE
 7x
Add:
3
 

 6x 2 11x  13  19x3 11x 2  7 x 17 .
SOLUTION
 7x
3
 

 6x 2 11x  13  19x3 11x 2  7 x 17
 7 x3  6 x 2  11x  13  19x3  11x 2  7 x  17
Remove parentheses

x 
19x  6
x
 11
x  11
x

7

17
7



x  13

Rearrange terms so that
like terms are adjacent
3
12x3
3
2

5x 2
2

4x

4
Combine like terms
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 5.1
Subtracting Polynomials
EXAMPLE
Subtract 5x 4 y 2  6x3 y  7 y  3x 4 y 2  5x3 y  6 y  8x.
SOLUTION
5x y
4
2
 
 6x3 y  7 y  3x 4 y 2  5x3 y  6 y  8x

5x4 y 2  6x3 y  7 y  3x 4 y 2  5x3 y  6 y  8x
Change subtraction to
addition and change the
sign of every term of the
polynomial in
parentheses.
5 x 4 y 2  3x 4 y 2  6 x 3 y  5 x 3 y  7 y  6 y  8x

  


Rearrange terms
2x4 y 2

11x 3 y

y
 8x
Combine like terms
Blitzer, Intermediate Algebra, 5e – Slide #16 Section 5.1