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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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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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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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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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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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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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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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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
Name ——————————————————————— Date ————————————
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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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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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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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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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