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Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 3 75
2.1 Evaluate and Graph Polynomial FunctionsGoal p Evaluate and graph polynomial functions.Georgia
PerformanceStandard(s)
MM3A1b, MM3A1c, MM3A1d
Your Notes
VOCABULARY
Polynomial
Polynomial function
Degree of a polynomial function
Leading coefficient
Standard form of a polynomial function
Synthetic substitution
End behavior
Even function
Odd function
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Your Notes
Decide whether the function is a polynomial function. If so, write it in standard form and state its degree and leading coefficient.
a. f (x) 5 3x3 1 4x2.5 2 6x2 b. f (x) 5 x2 1 3.7x 1 9x4
Solution
a. The function a polynomial function because the term has an exponent that is
.
b. The function a polynomial function written as in its standard form.
It has degree and a leading coefficient of .
Example 1 Identify polynomial functions
1. State the degree and leading coefficient of f (x) 5 22x3 1 2x2 2 3x4 1 5.
Checkpoint Complete the following exercise.
Use synthetic substitution to evaluate f(x) 5 2x4 1 3x3 2 6x2 1 3 when x 5 2.
Write the coefficients of f (x) in order of exponents. Write the value of x to the left. Bring down the leading coefficient. Multiply the leading coefficient by
and write the product under the second coefficient. . Multiply the previous sum by and write the
product under the third coefficient. Add. Repeat for all of the remaining coefficients.
2 3 26 0 3 coefficients
f (2) 5
Example 2 Evaluate by synthetic substitution
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Your NotesEND BEHAVIOR OF POLYNOMIAL FUNCTIONS
For the graph of f (x) 5 anxn 1 an 2 1xn 2 1 1 . . . 1 a1x 1 a0:
• If n is odd and an > 0, then f (x) → as x → 1`and f (x) → as x → 2`.
• If n is odd and an < 0, then f (x) → as x → 1`and f (x) → as x → 2`.
• If n is even and an > 0, then f (x) → as x → 1`and f (x) → as x → 2`.
• If n is even and an < 0, then f(x) → asx → 1` and f (x) → as x → 2`.
Example 3 Graph and analyze a polynomial function
(a) Graph the function f(x) 5 2x3 1 2x2 1 2x 2 1, (b) find the domain and the range of the function, (c) describe the degree and leading coefficient of the function, and (d) decide whether the function is even, odd, or neither and describe any symmetries in the graph.
Solutiona. Make a table of values and plot
the corresponding points. Connect the points with a smooth curve.
x 21 0 1 2 3
f(x)
y
x
y
1
1
b. The domain is and the range is .
c. The degree is and the leading coefficient is .
d. The function is because
f(2x) 5 2(2x)3 1 2(2x)2 1 2(2x) 2 1
5
which is not equal or . The graph has .
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Your Notes
Homework
2. Evaluate f (x) for x 5 22 using synthetic substitution.
3. Graph f(x).
y
x
y
1
1
Checkpoint Complete the following exercises using the function f(x) 5 2x4 1 3x3 1 x2 2 4x 2 1.
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Name ——————————————————————— Date ————————————
LESSON
2.1 PracticeDecide whether the function is a polynomial function. If it is, write the function in standard form.
1. f (x) 5 5x 1 2 2. f (x) 5 2x
3. g(x) 5 15 1 3x2 1 x 4. h(x) 5 1 }
2 x4 2 x2 1 3x3
State the degree and leading coeffi cient of the polynomial.
5. g(x) 5 2x2 2 4x 1 9 6. h(x) 5 7 2 3x
7. f (x) 5 2 3 } 4 x3 1 2x4 1 7 8. g(x) 5 8x 2 6 1 x3 Ï
}
2
Use direct substitution to evaluate the polynomial function for the given value of x.
9. f (x) 5 3x3 1 4x2 2 5x 1 7; x 5 1 10. g(x) 5 x5 2 2x 1 3x2 2 9; x 5 2
11. h(x) 5 4x 2 8x6 1 1; x 5 0 12. f (x) 5 x 1 2x3 2 x4 2 10; x 5 22
Use synthetic substitution to evaluate the polynomial function for the given value of x.
13. g(x) 5 2x3 2 5x2 1 4x 2 1; x 5 1 14. h(x) 5 x4 1 7x3 1 x2 2 2x 2 6; x 5 23
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Use what you know about end behavior to match the polynomial function with its graph.
15. f (x) 5 2x4 1 2x 2 1 16. f (x) 5 2x2 1 3x 2 2 17. f (x) 5 2x3 1 x2 2 1
A.
x
y
1
1
B.
x
y
1
1
C.
x
y
1
1
Decide whether the function is even, odd, or neither. Describe any symmetries in the graph.
18. f (x) 5 x3 19. f (x) 5 2x4 20. g(x) 5 x3 1 1
21. Business The cost of manufacturing a product can be
0 10 205 15 25 n0
250
500
750
10001250
1500
1750C
Units (thousands)
Co
st
(do
llars
)
modeled by the function C(n) 5 0.2n3 2 7n2 1 108n 1 100 where C is the cost in dollars and n is the number of units of the product in thousands.
a. State the degree of the function.
b. Complete the table of values for the function.
n 0 5 10 15 20
C
c. Use your table to graph the function.
LESSON
2.1 Practice continued
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2.2 Translate Graphs of Polynomial Functions
GeorgiaPerformanceStandard(s)
MM3A1a, MM3A1c, MM3A1d
Your Notes
Goal p Graph translations of polynomial functions.
1. Graph g(x) 5 x4 2 1. Compare the graph with the graph of f(x) 5 x4.
x
y
1
1
f
Checkpoint Complete the following exercise.
Example 1 Translate a polynomial function vertically
Graph g(x) 5 x4 1 1. Compare the graph with the graph of f(x) 5 x4.
1. Make a table of values and plot thecorresponding points.
x 22 21 0 1 2
y x
y
1
1
f
2. Connect the points with a smooth curve and check the end behavior. The degree is and the leading coefficient is . So, g(x) → as x → and g(x) → as x → .
3. Compare with f(x) 5 x4. The graph of g(x) 5 x4 1 1 is the graph of f(x) 5 x4 translated up unit. The domains of f and g are . The range of f is y ≥ 0 and the range of g is .The function f has x- and y-intercepts of 0 and g has a y-intercept of . Notice that both f and g are symmetric with respect to the and are functions because f(2x) 5 (2x)4 5 and g(2x) 5 (2x)4 1 1 5 .
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Your Notes Example 2 Translate a polynomial function horizontally
Graph g(x) 51}2(x 1 2)3. Compare the graph with the
graph of f(x) 51}2x3.
1. Make a table of values and plot thecorresponding points.
x 24 22 0 2 4
y
x
y
2
2
f
2. Connect the points with a smooth curve and check the end behavior. The degree is and the leading coefficient is . So, g(x) → as x → and g(x) → as x → .
3. Compare with f(x) 51}2x3. The graph of
g(x) 51}2 (x 1 2)3 is the graph of f(x) 5
1}2 x3 translated
to the left units. The domains and ranges of f and g are . The function f has x- and y-intercepts of 0 and g has an x-intercept of and a y-intercept of . Notice that f is symmetric with respect to the and is an function
because f(2x) 51}2(2x)3 5 . Notice that g
is because g(2x) 5
1}2(2x 1 2)3, which is not equal to or .
2. Graph g(x) 5 2(x 2 3)3. Compare the graph with the graph of f(x) 5 2x3.
2
2
f
x
y
2
2
f
Checkpoint Complete the following exercise.
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Your Notes
3. Graph g(x) 5 2(x 1 4)3 2 1. Compare the graph with the graph of f(x) 5 2x3.
x
y
1
21
f
Checkpoint Complete the following exercise.
Homework
Example 3 Translate a polynomial function
Graph g(x) 5 2(x 2 2)4 1 3. Compare the graph with the graph of f(x) 5 2x4.
1. Make a table of values and plot thecorresponding points.
x 0 1 2 3 4
y
x
y
1
2
f
2. Connect the points with a smooth curve and check the end behavior. The degree is and the leading coefficient is . So, g(x) → as x → and g(x) → as x → .
3. Compare with f(x) 5 2x4. The graph of g(x) 5 2 (x 2 2)4 1 3 is the graph of f(x) 5 2x4
translated units and .The domains of f and g are . The range of f is and the range of g is .The function f has x- and y-intercepts of 0 and g has x-intercepts of about and and a y-intercept of . Notice that f is symmetric with respect to the and is an function because f(2x) 5 2(2x)4 5 . Notice that gis because g(2x) 5 2(2x 2 2)4 1 3, which is not equal to
or .
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Match the function with its graph.
1. y 5 x3 1 1 2. y 5 x3 3. y 5 (x 1 1)3
A.
x
y
1
1
B.
x
y
1
1
C.
x
y
2
1
Explain how the graphs of f and g are related.
4. f (x) 5 x3, g(x) 5 x3 1 2 5. f (x) 5 x4, g(x) 5 (x 2 2)4
6. f (x) 5 2x3, g(x) 5 2x3 2 4 7. f (x) 5 2 x4, g(x) 5 2 (x 1 3)4
Graph the function. Compare the graph with the graph of f (x) 5 x3.
8. g(x) 5 (x 2 1)3 9. g(x) 5 x3 1 2
x
y
1
1
f
x
y
1
1
f
LESSON
2.2 Practice
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LESSON
2.2 Practice continued
Graph the function. Compare the graph with the graph of f (x) 5 x4.
10. g(x) 5 x4 2 4 11. g(x) 5 (x 1 2)4
x
y
1
1
f
x
y
1
1
f
12. Geometry The volume of a cube with side length x inches is given by V1 5 x3.
The volume of a cube with side length (x 1 1) inches is given by V2 5 (x 1 1)3.
a. Copy and complete the table.
x 1 2 3 4 5 6
V1
V2
b. Use the table from part (a) to graph V1 and V2. Explain how the graphs are related.
0 2 41 3 5 6 x0
50
100
150
200250
300
350V
Side length (inches)
Vo
lum
e (
cu
bic
in
ch
es)
c. Find the volume of each cube when x 5 8.
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2.3 Factor and Solve Polynomial EquationsGoal p Factor and solve polynomial equations.Georgia
PerformanceStandard(s)
MM3A3b, MM3A3d
Your Notes
VOCABULARY
Factored completely
Factor by grouping
Quadratic form
SPECIAL FACTORING PATTERNS
Sum of Two Cubes
a3 1 b3 5 (a 1 b)(a2 2 ab 1 b2)Example
x3 1 8 5 (x 1 2)( )Difference of Two Cubes
a3 2 b3 5 (a 2 b)(a2 1 ab 1 b2)Example
8x3 2 1 5 (2x 2 1)( )
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Your Notes
Factor the polynomial completely.
a. z3 2 125 5 z3 2 Difference of two cubes
5 (z 2 )( )
b. 81y4 1 192y 5 3y( ) Factor common monomial.
5 3y[ 1 ] Sum of two cubes
5 3y( )( )
Example 1 Factor the sum or difference of two cubes
1. Factor the polynomial 8x3 1 64 completely.
Checkpoint Complete the following exercise.
Factor the polynomial x3 2 2x2 2 9x 1 18 completely.
x3 2 2x2 2 9x 1 18 5 x2( ) 2 9( ) Factor by grouping.
5 Distributive property
5 Difference of two squares
Example 2 Factor by grouping
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Your Notes
Factor completely: (a) 16x4 2 256 and (b) 3y7 2 15y5 1 18y3.
a. 16x4 2 256 5 ( )2 2 2
5
5
b. 3y7 2 15y5 1 18y3 5 3y3( )
5
Example 3 Factor polynomials in quadratic form
2. x3 1 2x2 2 25x 2 50 3. x4 2 14x2 1 45
Checkpoint Factor the polynomial completely.
Find the real-number solutions of the equation x4 1 9 5 10x2.
x4 1 9 5 10x2 Write original equation.
5 0 Write in standard form.
5 0 Factor trinomial.
5 0 Difference of two squares
x 5 , x 5 , x 5 , x 5 Zero product property
The solutions are .
Example 4 Find real-number solutions
4. 2x5 1 24x 5 14x3
Checkpoint Find the real-number solutions.Homework
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LESSON
2.3 PracticeFind the greatest common factor of the terms in the polynomial.
1. 4x4 1 12x3 2. 10y2 1 4y 2 64 3. 16x5 2 8x
4. 32n5 2 64n3 1 16n2 5. 15p6 2 5p4 2 10p2 6. 36c9 1 13
Match the polynomial with its factorization.
7. 3x2 1 11x 1 6 A. 2x3(x 1 2)(x 2 2)(x2 1 3)
8. x3 2 4x2 1 4x 2 16 B. 2x(x 1 4)(x 2 4)
9. 125x3 2 216 C. (3x 1 2)(x 1 3)
10. 2x7 2 2x5 2 24x3 D. (x2 1 4)(x 2 4)
11. 2x5 1 4x4 2 4x3 2 8x2 E. 2x2(x2 2 2)(x 1 2)
12. 2x3 2 32x F. (5x 2 6)(25x2 1 30x 1 36)
Factor the sum or difference of cubes.
13. s3 2 1 14. q3 1 1
15. x3 2 27 16. a3 1 125
17. h3 1 64 18. 8y3 2 125
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Factor the polynomial by grouping.
19. x3 1 2x2 1 3x 1 6 20. z3 2 z2 1 5z 2 5
21. f 3 1 4f 2 1 f 1 4 22. m3 2 2m2 1 4m 2 8
23. 2x4 2 x3 1 6x 2 3 24. t3 2 2t2 2 9t 1 18
Find the real-number solutions of the equation.
25. w2 2 3w 5 0 26. v3 1 5v2 5 0
27. x2 2 5x 1 6 5 0 28. d2 2 16 5 0
29. 10s3 5 30s2 30. x3 1 x2 2 9x 2 9 5 0
Match the equation for volume with the appropriate solid.
31. V 5 x3 2 4x 32. V 5 x3 2 4x2 1 4x 33. V 5 x4 2 16
A. x 2 2
x 2 2x
B.
x 2 2x 1 2
x2 1 4
C.
x 1 2
x 2 2x
LESSON
2.3 Practice continued
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2.4 Solve Polynomial InequalitiesGoal p Solve polynomial inequalities.Georgia
PerformanceStandard(s)
MM3A3c
Your Notes
VOCABULARY
Polynomial inequality
Intervals
INEQUALITY AND INTERVAL NOTATION
In the intervals below, the real numbers a and b are the endpoints of each interval.
Bounded intervals:
Inequality a ≤ x ≤ b a < x < b a ≤ x < b a < x ≤ b
Notation
Unbounded intervals:
Inequality x ≥ a x > a x ≤ b x < b
Notation
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Your Notes
Solve x3 2 x2 < 12x algebraically.
First, write and solve the equation obtained by replacing with .
x3 2 x2 5 12x Write equation that corresponds to original inequality.
x3 2 x2 2 5 0 Write in standard form.
5 0 Factor.
x 5 , x 5 , or x 5 Zero product property
The numbers 0, 4, and 23 are the of the inequality x3 2 x2 < 12x. Plot 0, 4, and 23 on a number line, using because the values do not satisfy the inequality. The critical x-values partition the number line into four intervals. Test an x-value in each interval to see if it satisfies the inequality.
2124 2223 0 1 2 3 4 5 6 7 8
Test x 5 5:53 2 52 5 ,
Test x 5 21:(21)3 2 (21)2 5 ,
Test x 5 1:13 2 12 5 ,
Test x 5 24:(24)3 2 (24)2 5 ,
The solution set consists of all real numbers in the intervals and .
Example 1 Solve a polynomial inequality algebraically
1. Solve x3 2 16x > 0 algebraically.
Checkpoint Complete the following exercise.
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Your Notes
Solve 2x3 1 7x2 2 4x ≥ 0 by graphing.
The solution consists of the x-values for which the graph of y 5 2x3 1 7x2 2 4x lies or the x-axis.Find the graph's x-intercepts by letting y 5 0 and solve for x.
2x3 1 7x2 2 4x 5 0 Set y equal to 0.
5 0 Factor.
x 5 , x 5 , or x 5 Zero product property
Graph the polynomial and plot the
x
y
10
1
the x-intercepts , , and .
The graph lies on or above the x-axisbetween (and including) x 5and x 5 and to the right of
(and including) x 5 . The solution set consists of all
real numbers in the intervals and .
Example 2 Solve a polynomial inequality by graphing
2. Solve x4 2 4x2 ≤ 0 by graphing.
Checkpoint Complete the following exercise.
Homework
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Represent the inequality using interval notation.
1. x > 8 2. x ≤ 0 3. 4 ≤ x < 6
Represent the inequality using inequality notation.
4. [22, 3] 5. (21, 1) 6. [5, 1`)
Tell whether the given x-value is a solution of the inequality.
7. x3 2 4x < 0; x 5 1 8. x3 2 5x2 1 4x ≤ 0; x 5 1 } 2 9. x4 2 8 > 0; x 5 5
10. x3 2 x2 ≥ 6x; x 5 23 11. x3 1 4x2 ≥ 23x; x 5 4 12. x4 2 3x2 < 22; x 5 0
Solve the inequality algebraically.
13. x3 2 1 ≥ 0 14. x3 1 x2 2 6x ≤ 0 15. x4 2 1 < 0
16. x3 1 3x2 > 10x 17. 3x3 1 x2 ≥ 2x 18. x4 2 10x2 < 29
LESSON
2.4 Practice
Name ——————————————————————— Date ————————————
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LESSON
2.4 Practice continued
Use the graph of the corresponding equation to determine the solution set of the inequality.
19. x3 2 x2 2 2x ≤ 0 20. x3 2 2x2 2 8x > 0 21. x4 2 3x2 2 4 < 0
x
y
1
1
y 5 x3 2 x2 2 2x
x
y
5
1
y 5 x3 2 2x2 2 8x
x
y
5
1
y 5 x4 2 3x2 2 4
Solve the inequality using a graph.
22. x3 2 4x2 1 4x ≤ 0 23. 2x3 1 3x2 2 2x > 0 24. x4 2 8 ≥ 0
x
y
1
1
x
y
1
1
x
y
2
1
25. Geometry The volume V of a cube with a side length of x inches is greater than 27 cubic inches.
a. Copy and complete the table to determine the possible side lengths of the cube.
x 1 2 3 4 5 6
V 5 x3
b. Check the solution you found in part (a) by solving the inequality x3 > 27 algebraically.
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2.5 Apply the Remainder and Factor TheoremsGoal p Use theorems to factor polynomials.Georgia
PerformanceStandard(s)
MM3A3a
Your Notes
VOCABULARY
Polynomial long division
Synthetic division
Divide f(x) 5 4x4 1 5x2 2 9x 1 18 by x2 1 2x 1 4.
Write polynomial division in the same format you use when dividing numbers. Include a “0” as the coefficient of x3. At each stage, divide the term with the highest power in what is left of the dividend by the first term of the divisor. This gives the next term of the quotient.
x2 1 2x 1 4qwww4x4 1 0x3 1 5x2 2 9x 1 18
Write the result:
4x4 1 5x2 2 9x 1 18}}
x2 1 2x 1 45
Example 1 Use polynomial long division
You can check the result of a division problem by multiplying the quotient by the divisor and adding the remainder. The result should be the dividend.
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Your NotesREMAINDER THEOREM
If a polynomial f (x) is divided by x 2 k, then the remainder is r 5 .
Divide f(x) 5 x3 1 4x2 2 5x 1 3 by x 1 2.
Solution22 1 4 25 3
x3 1 4x2 2 5x 1 3}}
x 1 2 5
Example 2 Use synthetic division
FACTOR THEOREM
A polynomial f (x) has a factor x 2 k, if and only if f (k) 5 .
Factor f(x) 5 2x3 2 11x2 1 3x 1 36 completely given that x 2 3 is a factor.
Solution
Because x 2 3 is a factor of f(x), you know that f(3) 5 .Use synthetic division to find the other factors.
2 211 3 36
Use the result to write f(x) as a product of two factors and then factor completely.
f (x) 5 2x3 2 11x 1 3x 1 36 5 ( )( ) 5 ( )( )( )
Example 3 Factor a polynomial
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Your Notes
Find the other zeros of f(x) 5 x3 1 4x2 2 15x 2 18 given that f(21) 5 0.
Solution
Because f (21) 5 0, is a factor of f. Use synthetic division to find the other factors.
1 4 215 218
Use the result to write f(x) as a product of two factors and then factor completely.
f (x) 5 x3 1 4x2 2 15x 2 18
5 ( )( )
5 ( )( )( )
The zeros are .
Example 4 Finding zeros of functions
1. Use long division to divide x3 2 6x2 1 9 by x 2 4.
2. Use synthetic division to divide 2x3 1 4x2 2 3x 2 6 by x 1 3.
3. Factor f (x) 5 3x3 1 8x2 1 3x 2 2 given that x 1 2 is a factor.
4. Find the other zeros of f (x) 5 x3 2 4x2 2 11x 1 30 given that f(2) 5 0.
Checkpoint Complete the following exercises.
Homework
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LESSON
2.5 PracticeWrite the divisor, dividend, quotient, and remainder represented by the synthetic division.
1. 22 2 1 21 10
24 6 210
2 23 5 0
2. 4 3 210 0 25
12 8 32
3 2 8 27
Divide using polynomial long division.
3. (x2 2 6x 1 10) 4 (x 2 1) 4. (x2 1 2x 2 11) 4 (x 1 2)
5. (x2 1 3x 2 18) 4 (x 1 6) 6. (x2 1 3x 2 6) 4 (x 1 5)
7. (4x2 2 7x 2 4) 4 (x 2 4) 8. (2x2 2 x 1 5) 4 (x 1 3)
9. (x2 1 4) 4 (x 2 2) 10. (x3 1 11x2 1 25x 2 21) 4 (x 1 7)
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Divide using synthetic division.
11. (x2 1 8x 2 9) 4 (x 1 9) 12. (x2 2 x 2 1) 4 (x 1 1)
13. (x2 1 3x 2 10) 4 (x 2 2) 14. (x2 2 6x 1 4) 4 (x 1 3)
15. (x2 1 5x 2 7) 4 (x 1 4) 16. (2x2 2 7x 2 15) 4 (x 2 5)
17. (x3 1 x 1 2) 4 (x 2 1) 18. (x2 2 7) 4 (x 1 2)
You are given an expression for the area of the rectangle. Find an expression for the missing dimension.
19. A 5 x2 1 10x 1 21 20. A 5 x2 1 2x 2 8 21. A 5 x2 1 8x 1 15
x 1 3
?
x 1 4
?
x 1 5
?
22. Publishing The profi t P (in thousands of dollars) for an educational publisher can be modeled by P 5 2b3 1 5b2 1 b where b is the number of workbooks printed (in thousands). Currently, the publisher prints 5000 workbooks and makes a profi t of $5000. What lesser number of workbooks could the publisher print and still yield the same profi t?
LESSON
2.5 Practice continued
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2.6 Find Rational ZerosGoal p Find all real zeros of a polynomial function.Georgia
PerformanceStandard(s)
MM3A3a, MM3A3d
Your Notes
THE RATIONAL ROOT THEOREM
If f(x) 5 an x n 1 . . . 1 a1x 1 a0 has coefficients, then every rational zero of f has the following form:
p}q 5
factor of constant term }}}factor of leading coefficient
Find all real zeros of f(x) 5 x3 2 4x2 2 7x 1 10.
1. List the possible rational zeros. The leading coefficient is and the constant term is . So, the possible
rational zeros are: x 5 , , ,
2. Test these zeros using synthetic division. Test x 5 :
1 24 27 10
is a zero.
3. Factor the trinomial and use the factor theorem.
f (x) 5 ( )( )
5
The zeros of f are .
Example 1 Find zeros when the leading coefficient is 1
1. f (x) 5 x3 1 3x2 2 10x 2 24
Checkpoint Find all real zeros of the function.
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Your Notes Example 2 Find zeros when the leading coefficient is not 1
Find all real zeros of f(x) 5 8x4 1 2x3 2 21x2 2 7x 1 3.
1. List the possible rational zeros of f:
2. Choose a reasonable value to check using the graph of the function.
3. Check x 5 :
8 2 221 27 3
is a zero. 4. Factor out a binomial.
f (x) 5 Write as a product of factors.
5 Factor out .
5 Multiplyby .
5. Repeat the steps above for g(x) 5 .Any zero of g will also be a zero of f. Synthetic division
shows that is a zero and yields the quotient
. Factoring a 4 out of the quotient yields f (x) 5 .
6. Find the remaining zeros by solving 5 0.
x 5 Use quadratic formula.
x 5 Simplify.
The real zeros of f are .
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Your Notes
Sandbox You are building a wooden square sandbox for a local playground. You want the volume of the box to be 16 cubic feet. You want the height of the box to be x feet and the length of each side of the square base to be x 1 3 feet. What are the dimensions?
1. Write an equation for the volume of the sandbox. The volume is V 5 Bh where B 5 base area and h 5 height.
Volume (cubic feet)
5Area of base (square feet)
pHeight(feet)
16 5 (x 1 3)2 p x
16 5 Write the equation.
16 5 Multiply.
0 5Subtract from each side.
2. List the possible rational solutions:
3. Test the possible rational solutions. Only positive x-values make sense.
1
4. Check for other solutions. The other possible rational solutions solutions, so x 5 is the solution. The height of the sandbox should be
foot and each side of the base should be 5 feet.
Example 3 Solve a multi-step problem
2. f (x) 5 9x4 1 12x3 2 26x2 2 11x 1 6
Checkpoint Find all real zeros of the function.Homework
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1. Can you use the rational root theorem to fi nd the zeros of the polynomial function f (x) 5 0.4x2 2 3x 1 2.2? Explain why or why not.
List the possible rational zeros of the function using the rational root theorem.
2. f (x) 5 x3 2 5x 1 16 3. g(x) 5 x4 1 8x2 2 18
4. h(x) 5 x5 1 2x4 2 3x 2 24 5. f (x) 5 x8 2 2x5 1 x4 2 3x 1 20
6. h(x) 5 2x3 2 5x2 2 9 7. g(x) 5 3x3 1 7x 1 12
Use synthetic division to decide which of the following are zeros of the function: 23, 21, 1, 3.
8. f (x) 5 x3 2 9x 1 3 9. g(x) 5 x4 1 3x3 2 7x2 2 27x 2 18
10. g(x) 5 2x4 2 9x3 1 8x2 1 9x 2 10 11. h(x) 5 3x4 1 5x2 2 17x 1 9
Find all rational zeros of the function.
12. f (x) 5 x3 1 x2 2 14x 2 24 13. f (x) 5 x3 2 2x2 2 x 1 2
x
y
10
2
x
y
1
1
LESSON
2.6 Practice
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LESSON
2.6 Practice continued
Find all real zeros of the function.
14. f (x) 5 x3 2 8x2 2 23x 1 30 15. g(x) 5 x3 1 2x2 2 11x 2 12
16. h(x) 5 x3 2 7x2 1 2x 1 40 17. h(x) 5 x3 1 9x2 2 4x 2 36
18. g(x) 5 x4 2 5x3 1 7x2 1 3x 2 10 19. f (x) 5 x4 2 2x3 2 7x2 1 8x 1 12
20. f (x) 5 x4 1 3x3 2 21x2 2 43x 1 60 21. g(x) 5 x4 1 x3 2 11x2 2 9x 1 18
22. Crafts You have 18 cubic inches of wax, and you
x
x 1 3
want to make a candle in the shape of a pyramid with a square base as shown.
a. Write an equation that shows that the volume of the candle is 18 cubic inches.
b. Use the rational root theorem to list all possible rational solutions of the equation in part (a).
c. Find all real values of x that are valid as a dimension of the candle.
d. Find the dimensions of the candle.
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2.7 Apply the Fundamental Theorem of AlgebraGoal p Classify the zeros of polynomial functions.Georgia
PerformanceStandard(s)
MM3A3a
Your Notes
VOCABULARY
Repeated solution
THE FUNDAMENTAL THEOREM OF ALGEBRA
Theorem: If f (x) is a polynomial of degree n where n 0, then the equation f (x) 5 0 has at least solution in the set of complex numbers.
Corollary: If f (x) is a polynomial of degree n wheren 0, then the equation f (x) 5 0 has exactly
solutions provided each solution repeated twice is counted as solutions, each solution repeated three times is counted as solutions, and so on.
Find the number of solutions or zeros for each equation or function.
a. Because x3 2 3x2 1 9x 2 27 5 0 is a degree polynomial equation, it has solutions.
b. Because f (x) 5 x4 1 6x3 2 32x is a degree polynomial function, it has zeros.
Example 1 Find the number of solutions or zeros
1. State the number of zeros of
f (x) 5 x3 2 2x2 2 9x 1 18.
Checkpoint Complete the following exercise.
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Your Notes
Find all zeros of f(x) 5 x5 2 5x4 2 9x3 2 5x2 2 8x 1 12.
Solution1. Find the rational zeros of f. Because f is a fifth-degree
function, it has zeros. The possible rational zeros are . Using synthetic division, you can determine that is a zero repeated twice and is also a zero.
2. Write f (x) in factored form. Dividing f by its known factors gives a quotient of . So,
f (x) 5 .
3. Find the complex zeros of f. Use the quadratic formula to factor the trinomial into linear factors.
f (x) 5
The zeros of f are .
Example 2 Find the zeros of a polynomial function
2. f (x) 5 x4 2 7x3 1 13x2 1 x 2 20
Checkpoint Find all zeros of the polynomial function.
COMPLEX CONJUGATES THEOREM
If f is a polynomial function with coefficients, and is an imaginary zero of f, then is also a zero of f.
IRRATIONAL CONJUGATES THEOREM
Suppose f is a polynomial function with coefficients, and a and b are rational numbers such that Ï
}
b is irrational. If is a zero of f, then is also a zero of f.
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Your Notes
Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and 22 and 3 1 i as zeros.
Because the coefficients are real and 3 1 i is a zero, must also be a zero. Use the three zeros and the
factor theorem to write f (x) as a product of three factors.
f (x) 5 ( )[x 2 ( )][x 2 ( )] Factored form
5 ( )[ ][ ] Regroup terms.
5 Multiply.
5 Expand, usei2 5 21.
5 Simplify.
5 Multiply.
5 Combinelike terms.
Example 3 Use zeros to write a polynomial function
3. Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and 4 and 1 1 Ï
}
6 as zeros.
Checkpoint Complete the following exercise.
DESCARTES’ RULE OF SIGNS
Let f (x) 5 anxn 1 an 2 1xn 2 1 1 . . . 1 a2x2 1 a1x 1 a0be a polynomial function with real coefficients.
• The number of real zeros of f is equal to the number of changes in sign of the coefficients of or is less than this by an number.
• The number of real zeros of f is equal to the number of changes in sign of the coefficients of or is less than this by an number.
You can check this result by evaluating f (x) at each of its three zeros.
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Your Notes
Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for
f(x) 5 2x5 2 7x4 1 12x3 1 2x2 1 4x 1 6.
Solutionf(x) 5 2x5 2 7x4 1 12x3 1 2x2 1 4x 1 6
The coefficients of f (x) have sign changes, so f has positive real zero(s).
f (2x) 5 2(2x)5 2 7(2x)4 1 12(2x)3
1 2(2x)2 1 4(2x) 1 6
5
The coefficients of f (2x) have sign changes, so f has negative real zero(s).
Positive real zeros
Negative real zeros
Imaginary zeros
Total zeros
Example 4 Use Descartes’ rule of signs
4. Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for f (x) 5 3x5 2 4x4 1 x3 1 6x2 1 7x 2 8.
Checkpoint Complete the following exercise.
Homework
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Identify the number of solutions or zeros.
1. 2x2 1 5x 2 9 5 0 2. f (t) 5 t3 1 4t2 2 7
3. g(z) 5 2z3 1 z2 1 6z 2 3 4. x4 2 6x2 1 8x 2 12 5 0
5. h(x) 5 2x5 2 3x4 1 x 2 9 6. 26x4 1 7x2 2 15 5 0
7. f (y) 5 22y 1 3 8. 3r3 2 r2 1 5r 2 21 5 0
Given that f (x) has real coeffi cients and x 5 k is a zero, what other number must be a zero?
9. k 5 3 2 Ï}
2 10. k 5 i 11. k 5 22 2 9i
12. k 5 Ï}
3 2 i 13. k 5 1 1 Ï}
5 i 14. k 5 Ï}
2 1 Ï}
7 i
Find all the zeros of the polynomial function.
15. g(x) 5 x3 2 8x2 2 15x 1 54 16. f (x) 5 x4 2 3x3 2 17x2 1 39x 2 20
17. h(x) 5 x3 1 6x2 1 x 1 6 18. g(x) 5 x3 1 8x2 2 7x 2 56
19. f (x) 5 x3 2 2x2 1 9x 2 18 20. h(x) 5 x4 2 x2 2 20
LESSON
2.7 Practice
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LESSON
2.7 Practice continued
Write a polynomial function f of least degree that has rational coeffi cients, a leading coeffi cient of 1, and the given zeros.
21. 29 22. 25, 4
23. 23, 21 24. 21, 0, 1
25. 21, 2, 6 26. 22, 21, 10
27. The graph f (x) 5 x3 2 x2 2 8x 1 12 is shown at the
x
y
1
1
right. How many real zeros does the function have? How many imaginary zeros does the function have?
28. Geometry A square piece of sheet metal is
10 in.x in.
10 in.
x in.
10 inches by 10 inches. Squares of side length x are cut from the corners and the remaining piece is folded to make an open box. The volume of the box is modeled by V(x) 5 4x3 2 40x2 1 100x. What size square(s) can be cut from the corners to give a box with a volume of 25 cubic inches?
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2.8 Analyze Graphs of Polynomial FunctionsGoal p Use intercepts to graph polynomial functions.Georgia
PerformanceStandard(s)
MM3A1b, MM3A1d
Your Notes
VOCABULARY
Local maximum
Local minimum
Multiplicity of a root
MULTIPLICITY OF A ROOT
For the polynomial equation f(x) 5 0, k is a repeated solution, or a root with a ,if and only if the factor x 2 k has an exponent greater than when f(x) is factored completely. If the exponent is , the graph of f the x-axisat the zero. If the exponent is , the graph of f the x-axis at the zero.
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Your Notes
1. Graph the function y
x
y
1
1
f (x) 5 2(x 2 2)(x 1 1)(x 2 1).
Checkpoint Complete the following exercise.
TURNING POINTS OF POLYNOMIAL FUNCTIONS
The graph of every polynomial function of degree n has at most turning points.
Graph the function y
x
y
1
1
f(x) 51}4
(x 1 1)2(x 2 4).
1. Use the intercepts. Because and are zeros of f,
plot ( , ) and ( , ).
2. Plot points between and beyond the x-intercepts.
x 22 0 1 2 3
y
3. Determine the end behavior and multiplicity. Because f has factors of the form x 2 k, and a
constant factor of , it is a function with
a leading coefficient. So, f (x) → asx → 2` and f (x) → as x → 1`. The zero 21 is repeated . So, the graph of f the x-axis at (21, 0).
4. Draw the graph so that it passes through the plotted points and has the appropriate end behavior.
Example 1 Use x-intercepts to graph a polynomial function
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Your Notes
2. Use a graphing calculator to identify the x-intercepts, local maximums, and local minimums of the graph of f (x) 5 x4 1 x3 2 5x2 1 4.
Checkpoint Complete the following exercise.
Graph the function. Identify the x-intercepts and the points where the local maximums and local minimums occur.
a. f (x) 5 x3 2 4x2 1 6
b. f (x) 5 2x4 1 3x3 1 x2 2 4x
a. Use a graphing calculator to graph the function.Notice that the graph of f has
x-intercepts and turning points. Use the graphing calculator’s zero, maximum, and minimum features to approximate the coordinates of the points.
The x-intercepts of the graph are . The function has a local maximum at
( , ) and a local minimum at ( , ).
b. Use a graphing calculator to graph the function. Notice that the graph of f has
x-intercepts and turning points. Use the graphing calculator’s zero, maximum, and minimum features to approximate the coordinates of the points.
The x-intercepts of the graph are . The function
has local maximums at ( , ) and ( , ). The function has a local minimum at ( , ).
Example 2 Find turning points
Homework
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LESSON
2.8 Practice 1. True or False If k is a zero of the polynomial function f, then k is an x-intercept of
the graph of f (x). Explain your answer.
Determine the lowest-degree polynomial that has the given graph.
2. y
x2
2
3. y
x1
1
4. y
x2
2
Estimate the coordinates of each turning point and state whether each corresponds to a local maximum or a local minimum.
5. y
x2
2
6. y
x1
1
7. y
x1
1
Match the graph with its function.
8. f (x) 5 2x4 2 3x2 2 2 9. f(x) 5 2x6 2 6x4 1 4x2 2 2 10. f (x) 5 22x4 1 3x2 2 2
A.
x
y
1
1
B.
x
y
1
1
C.
x
y
1
1
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Determine the x-intercepts of the function.
11. g(x) 5 (x 1 4)(x 2 1) 12. h(x) 5 (x 2 2)(x 2 3)
13. f (x) 5 x(x 1 4)(x 2 5) 14. f (x) 5 (x 1 3)(x 1 1)(x 2 8)
15. g(x) 5 (x 1 6)2 16. h(x) 5 (x 2 1)(x 2 7)2
Graph the function.
17. f (x) 5 (x 1 1)(x 2 2) 18. g(x) 5 (x 2 3)(x 2 1) 19. h(x) 5 (x 1 6)(x 1 7)
x
y
1
1
x
y
1
1
x
y
2
22
20. h(x) 5 0.9(x 1 5)(x 2 2) 21. g(x) 5 (x 2 3)2 22. f (x) 5 (x 1 1)(x 2 1)(x 2 3)
x
y
222
x
y
1
1
x
y
1
1
LESSON
2.8 Practice continued
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LESSON
2.8 Practice continued
23. Let f be a fourth-degree polynomial function with the zeros 22, 6, 2i, and 22i.
a. How many distinct linear factors does f (x) have?
b. How many distinct solutions does f (x) 5 0 have?
c. What are the x-intercepts of the graph of f ?
24. Manufacturing You are designing an open box from a piece of cardboard that is 18 inches by 18 inches. Squares of side length x are cut from the corners and the remaining piece is folded to make an open box. The volume of the box is given by the function
V 5 4x3 2 72x2 1 324x.
Using a graphing calculator, you would obtain the graph shown below.
a. What is the domain of the volume function? Explain.
b. Use the graph to estimate the length of the cut x that will maximize the volume of the box.
c. Estimate the maximum volume the box can have.
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Words to ReviewGive an example of the vocabulary word.
Polynomial
Degree of a polynomial function
Standard form of a polynomial function
End behavior
Odd function
Polynomial function
Leading coefficient
Synthetic substitution
Even function
Factored completely
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Factor by grouping
Polynomial inequality
Polynomial long division
Quadratic form
Intervals
Synthetic division
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Repeated solution
Local minimum
Local maximum
Multiplicity of a root
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