Download Chapter Review Exercises

Chapter 4 Review Exercise
4-5
PARTIAL FRACTIONS
A rational function P(x)/D(x) often can be decomposed into a
sum of simpler rational functions called partial fractions. If
the degree of P(x) is less than the degree of D(x), then P(x)/D(x)
is called a proper fraction. We have the following important
theorems:
Equal Polynomials. Two polynomials are equal to each other if
and only if the coefficients of terms of like degree are equal.
Linear and Quadratic Factor Theorem. A polynomial with
real coefficients can be factored into a product of linear
and/or quadratic factors with real coefficients where the
linear and quadratic factors are prime relative to the real
numbers.
Partial Fraction Decomposition. Any proper fraction
P(x)/D(x) reduced to lowest terms can be decomposed into
the sum of partial fractions as follows:
1. If D(x) has a nonrepeating linear factor of the form ax b,
then the partial fraction decomposition of P(x)/D(x) contains
a term of the form
A
ax b
349
2. If D(x) has a k-repeating linear factor of the form (ax b)k,
then the partial fraction decomposition of P(x)/D(x) contains
k terms of the form
A1
Ak
A2
2
ax b (ax b)
(ax b)k
A1, A2, . . . , Ak constants
3. If D(x) has a nonrepeating quadratic factor of the form ax2 bx c, which is prime relative to the real numbers, then the
partial fraction decomposition of P(x)/D(x) contains a term
of the form
Ax B
ax2 bx c
A, B constants
4. If D(x) has a k-repeating quadratic factor of the form (ax2 bx c)k, where ax2 bx c is prime relative to the real
numbers, then the partial fraction decomposition of
P(x)/D(x) contains k terms of the form
A2 x B2
Ak x Bk
A1x B1
2
2
2
2
ax bx c (ax bx c)
(ax bx c)k
A a constant
A1, . . . , Ak,
B1, . . . , Bk constants
Chapter 4 Review Exercise
Work through all the problems in this chapter review, and
check answers in the back of the book. Answers to all review
problems are there, and following each answer is a number
in italics indicating the section in which that type of problem
is discussed. Where weaknesses show up, review appropriate
sections in the text.
A
1. Use synthetic division to divide P(x) 2x3 3x2 1 by
D(x) x 2, and write the answer in the form P(x) D(x)Q(x) R.
2. If P(x) x5 4x4 9x2 8, find P(3) using the remainder
theorem and synthetic division.
3. What are the zeros of P(x) 3(x 2)(x 4)(x 1)?
4. If P(x) x2 2x 2 and P(1 i) 0, find another zero of
P(x).
5. Let P(x) be the polynomial whose graph is shown in the
figure.
(A) Assuming that P(x) has integer zeros and leading coefficient 1, find the lowest-degree equation that could produce this graph.
(B) Describe the left and right behavior of P(x).
P (x)
5
5
5
x
5
6. According to the upper and lower bound theorem, which of
the following are upper or lower bounds of the zeros of
P(x) x3 4x2 2?
2, 1, 3, 4
7. How do you know that P(x) 2x3 3x2 x 5 has at least
one real zero between 1 and 2?
8. Write the possible rational zeros for P(x) x3 4x2 x 6.
350
4 Polynomial and Rational Functions
9. Find all rational zeros for P(x) x3 4x2 x 6.
10. Find the domain and x intercept(s) for:
3x
2x 3
(A) f (x) (B) g(x) 2
x4
x x6
11. Find the horizontal and vertical asymptotes for the functions in Problem 10.
12. Decompose into partial fractions:
7x 11
(x 3)(x 2)
B
13. Let P(x) x3 3x2 3x 4.
(A) Graph P(x) and describe the graph verbally, including
the number of x intercepts, the number of turning
points, and the left and right behavior.
(B) Use the bisection method to approximate the largest x
intercept to one decimal place.
14. If P(x) 8x4 14x3 13x2 4x 7, find Q(x) and R such
that P(x) (x 14)Q(x) R. What is P( 41)?
15. If P(x) 4x3 8x2 3x 3, find P(12) using the remainder theorem and synthetic division.
26. Decompose into partial fractions:
x2 3x 4
x(x 2)2
27. Decompose into partial fractions:
8x2 10x 9
2x3 3x2 3x
C
28. Use synthetic division to divide P(x) x3 3x 2 by
[x (1 i)], and write the answer in the form
P(x) D(x)Q(x) R.
29. Find a polynomial of lowest degree with leading coefficient
1 that has zeros 12 (multiplicity 2), 3, and 1 (multiplicity
3). (Leave the answer in factored form.) What is the degree
of the polynomial?
30. Repeat Problem 29 for a polynomial P(x) with zeros 5,
2 3i, and 2 3i.
31. Find all zeros (rational, irrational, and imaginary) exactly
for P(x) 2x5 5x4 8x3 21x2 4.
32. Factor the polynomial in Problem 31 into linear factors.
33. Solve
4x2 4x 3
0
2x 3x2 11x 6
16. Use the quadratic formula and the factor theorem to factor
P(x) x2 2x 1.
17. Is x 1 a factor of P(x) 9x26 11x17 8x11 5x4 7?
Explain, without dividing or using synthetic division.
18. Determine all rational zeros of P(x) 2x3 3x2 18x 8.
19. Factor the polynomial in Problem 18 into linear factors.
20. Find all rational zeros of P(x) x3 3x2 5.
21. Find all zeros (rational, irrational, and imaginary) exactly
for P(x) 2x4 x3 2x 1.
22. Factor the polynomial in Problem 21 into linear factors.
23. Solve 2x3 3x2 11x 6. Write the answer in inequality
and interval notation.
24. Let P(x) x4 2x3 30x2 25.
(A) Find the smallest positive and the largest negative integers that, by Theorem 2 in Section 4-3, are upper and
lower bounds, respectively, for the real zeros of P(x).
(B) Use the bisection method to approximate the largest
real zero of P(x) to two decimal places.
(C) Use a graphing utility to approximate the real zeros of
P(x) to two decimal places.
x1
25. Let f(x) 2x 2
(A) Find the domain and the intercepts for f.
(B) Find the vertical and horizontal asymptotes for f.
(C) Sketch a graph of f. Draw vertical and horizontal
asymptotes with dashed lines.
3
Write the answer in both inequality and interval notation.
34. What is the minimal degree of a polynomial P(x), given that
P(1) 4, P(0) 2, P(1) 5, and P(2) 3? Justify
your conclusion.
35. If P(x) is a cubic polynomial with integer coefficients and if
1 2i is a zero of P(x), can P(x) have an irrational zero?
Explain.
36. The solutions to the equation x3 27 0 are the cube roots
of 27.
(A) How many cube roots of 27 are there?
(B) 3 is obviously a cube root of 27; find all others.
37. Let P(x) x4 2x3 500x2 4,000.
(A) Find the smallest positive integer multiple of 10 and
the largest negative integer multiple of 10 that, by Theorem 2 in Section 4-3, are upper and lower bounds, respectively, for the real zeros of P(x).
(B) Approximate the real zeros of P(x) to two decimal
places.
38. Graph
f (x) x2 2x 3
x1
Indicate any vertical, horizontal, or oblique asymptotes
with dashed lines.
Chapter 4 Review Exercise
39. Use a graphing utility to find any horizontal asymptotes for
f (x) 351
42. Construction. A grain silo is formed by attaching a hemisphere to the top of a right circular cylinder (see the figure).
If the cylinder is 18 feet high and the volume of the silo is
486 cubic feet, find the common radius of the cylinder
and the hemisphere.
2x
x2 3x 4
40. Decompose into partial fractions:
5x2 2x 9
x 3x3 x2 3x
x
4
x
APPLICATIONS
18 feet
Express the solutions to each problem as the roots of a
polynomial equation of the form P(x) 0. Find rational
solutions exactly and irrational solutions to one decimal
place. Use a graphing utility or bisection only if necessary.
41. Architecture. An entryway is formed by placing a rectangular door inside an arch in the shape of the parabola with
graph y 16 x2, x and y in feet (see the figure). If the area
of the door is 48 square feet, find the dimensions of the
door.
★
y
16
43. Manufacturing. A box is to be made out of a piece of cardboard that measures 15 by 20 inches. Squares, x inches on a
side, will be cut from each corner, and then the ends and
sides will be folded up (see the figure). Find the value of x
that would result in a box with a volume of 300 cubic
inches.
y 16 x 2
20 in.
x
15 in.
x
★
4
x
4
44. Geometry. Find all points on the graph of y x2 that are 3
units from the point (1, 4).