1. Can any quotient of polynomials be decomposed into at least two partial fractions? If so, explain why, and if not, give an example of such a fraction
2. Can you explain why a partial fraction decomposition is unique? (Hint: Think about it as a system of equations.)
3. Can you explain how to verify a partial fraction decomposition graphically?
4. You are unsure if you correctly decomposed the partial fraction correctly. Explain how you could double-check your answer.
5. Once you have a system of equations generated by the partial fraction decomposition, can you explain another method to solve it? For example if you had
3x2+8x+157x+13=x+1A+3x+5B , we eventually simplify to
7x+13=A(3x+5)+B(x+1) . Explain how you could intelligently choose an
-value that will eliminate either
or
and solve for
and
.
For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.
6.
x2+10x+245x+16
7.
x2−5x−243x−79
8.
x2−2x−24−x−24
9.
x2+7x+1010x+47
10.
6x2+25x+25x
11.
20x2−13x+232x−11
12.
x2+7x+10x+1
13.
x2−95x
14.
x2−2510x
15.
x2−46x
16.
x2−6x+52x−3
17.
x2−x−64x−1
18.
x2+8x+154x+3
19.
x2−5x+63x−1
For the following exercises, find the decomposition of the partial fraction for the repeating linear factors.
20.
(x+4)2−5x−19
21.
(x−2)2x
22.
(x+3)27x+14
23.
(4x+5)2−24x−27
24.
(6x−7)2−24x−27
25.
(x−7)25−x
26.
2x2+12x+185x+14
27.
2x(x+1)25x2+20x+8
28.
5x(3x+5)24x2+55x+25
29.
2x2(3x+2)254x3+127x2+80x+16
30.
x2(x2+12x+36)x3−5x2+12x+144
For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.
31.
(x+2)(x2+x+3)4x2+6x+11
32.
(x−1)(x2+6x+11)4x2+9x+23
33.
(x−1)(x2+3x+8)−2x2+10x+4
34.
(x+1)(x2+5x−2)x2+3x+1
35.
(x+3)(x2+6x+1)4x2+17x−1
36.
(x+5)(x2+7x−5)4x2
37.
x3−14x2+5x+3
38.
x3+8−5x2+18x−4
39.
x3+273x2−7x+33
40.
x3−125x2+2x+40
41.
8x3−274x2+4x+12
42.
125x3−1−50x2+5x−3
43.
x4+216x−2x3−30x2+36x+216
For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor.
44.
(x2+4)23x3+2x2+14x+15
45.
(x2+1)2x3+6x2+5x+9
46.
(x2−3)2x3−x2+x−1
47.
(x+2)2x2+5x+5
48.
(x2+2x+9)2x3+2x2+4x
49.
(x2+3x+25)2x2+25
50.
(2x2+x+14)22x3+11x+7x+70
51.
x(x2+4)25x+2
52.
x(x2+6)2x4+x3+8x2+6x+36
53.
(x2−x)22x−9
54.
(x2+2x)25x3−2x+1
For the following exercises, find the partial fraction expansion.
55.
(x+1)3x2+4
56.
(x−2)3x3−4x2+5x+4
For the following exercises, perform the operation and then find the partial fraction decomposition.
57.
x+87+x−25−x2−6x−16x−1
58.
x−41−x+63−x2+2x−242x+7
59.
x2−162x−x2+6x+81−2x−x2−4xx−5
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