Chapter 5 Polynomials, Polynomial Functions, and Factoring
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Transcript of Chapter 5 Polynomials, Polynomial Functions, and Factoring
Chapter 5Polynomials, Polynomial Functions, and Factoring
§ 5.1
Introduction to Polynomials and Polynomial Functions
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 5.1
Polynomials
A polynomial is a single term or the sum of two or more terms containing variables with whole number exponents.
Consider the polynomial: 6523 34 xxxThis polynomial contains four terms. It is customary to write the terms inorder of descending powers of the variable. This is the standard form of apolynomial.
55 9 2
2
1 54 , or , 7 , 6 , , and 9.2 2 3
mx m z x zx
Terms
3 2 2 33 5, 4 5 8, and 5x m m p t s
1 2 13 , 9 , and x x xx
• Polynomials
• Not Polynomials
A polynomial is a term or a finite sum of terms in which all variables have whole number exponents and no variables appear in
denominators.
Polynomials
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 5.1
Polynomials
The degree of a polynomial is the greatest degree of any term of the polynomial. The degree of a term
6523 34 xxx
mn yax 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:
3 is the leading coefficient. The degree is 4.
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 5.1
PolynomialsCONTINUE
D
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 . Its coefficient, -5, is the leading coefficient.
735 yx
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 5.1
PolynomialsEXAMPLE
SOLUTION
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.
735 yx
Term Coefficient Degree (Sum of Exponents on the Variables)
12 4 + 1 = 5-5 3 + 7 = 10-1 2 + 0 = 2
4 4 0 + 0 = 0
4512 2734 xyxyx
2x
yx412
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 5.1
Polynomials
2524)( 23 xxxxf
is an example of a polynomial function. In a polynomialfunction, 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 #9 Section 5.1
PolynomialsEXAMPLE
SOLUTION
The polynomial function
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).
896,107575,572212 2 xxxf
To find f (8), we replace each occurrence of x in the function’s formula with 8.
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 5.1
Polynomials
896,107575,572212 2 xxxf
896,1078575,57822128 2 f
896,1078575,576422128 f
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.
CONTINUED
896,107600,460568,1418 f
928,4268 f
Original functionReplace each occurrenceof x with 8Evaluate exponentsMultiplyAdd
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 4.1
PolynomialsCheck Point
2
2 find
6534
function polynomial For the23
fxxxxf
16
62523242 23 f
6101232
Blitzer, Intermediate Algebra, 5e – Slide #12 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 #13 Section 5.1
Graphs of PolynomialsEXAMPLE
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
Smooth rounded curve
Discontinuous break
Sharp Corner
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 5.1
Adding Polynomials
EXAMPLE
SOLUTIONAdd: . 1771119131167 2323 xxxxxx
1771119131167 2323 xxxxxx
1771119131167 2323 xxxxxx Remove parentheses
4 4 5 12
1713711116197
23
2233
xxx
xxxxxx Rearrange terms so that like terms are adjacent
Combine like terms
Polynomials are added by removing the parentheses that surround each polynomial (if any) and then combining like terms.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 4.1
Adding Polynomials p 311
Check Point 6
1364
34723
23
xx
xx
10103 23 xx
Blitzer, Intermediate Algebra, 5e – Slide #16 Section 4.1
Adding PolynomialsCheck Point
7
91282
357 23
23
yxyxy
yxyxy
91539 23 yxyxy
Blitzer, Intermediate Algebra, 5e – Slide #17 Section 5.1
Subtracting Polynomials
EXAMPLE
SOLUTIONSubtract . 8653765 324324 xyyxyxyyxyx
xyyxyxyyxyx 8653765 324324
xyyxyxyyxyx 8653765 324324 Change subtraction to addition and change the sign of every term of the polynomial in parentheses.
Rearrange terms
Combine like terms
xyyxyx
xyyyxyxyxyx
8 11 2
8675635
324
332424
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 #18 Section 4.1
Subtracting PolynomialsCheck Point
8
)173(4 -
)95(14 23
23
xxx
xxx
108210 23 xxx
1734-
9514 23
23
xxx
xxx
DONE
Blitzer, Intermediate Algebra, 5e – Slide #20 Section 5.1
Graphs of Polynomials
The Leading Coefficient TestAs x increases or decreases without bound, the graph of a polynomial function eventually rises or falls. In particular,
Odd-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.
Even-Degree Polynomials
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 #21 Section 5.1
PolynomialsEXAMPLE
SOLUTION
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
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?
.5375.0 34 xxxf
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 #22 Section 5.1
Polynomials
The Degree of TIf , the degree of is n. The degree of a nonzero constant is 0. The constant 0 has no defined degree.
nax0a
nax
Adding PolynomialsPolynomials are added by removing the parentheses that surround each polynomial (if any) and then combining like terms.
Subtracting PolynomialsTo subtract two polynomials, change the sign of every term of the second polynomial. Add this result to the first polynomial.