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Partial Fractions Worksheet

Partial Fractions Worksheet
  • Page 1
 1.  
Find the single reduced fraction of (4x - 2 + 5x - 3).
a.
9xx2 - 5x + 6
b.
9x - 22x2 + 5x + 6
c.
9x + 22x2 - 5x + 6
d.
9x - 22x2 - 5x + 6


Solution:

(4x - 2 + 5x - 3) = 4 (x - 3) + 5 (x - 2)(x - 2) (x - 3)

= 9x - 22x2 - 5x + 6


Correct answer : (4)
 2.  
Choose the values of A & B such that x + 22(x + 4)(x -2) = Ax + 4 + Bx - 2.
a.
-3, 4
b.
4, 3
c.
3, 3
d.
3, -4


Solution:

x + 22(x + 4)(x -2) = Ax + 4 + Bx - 2

x + 22 = A(x - 2) + B(x + 4)
[Multiply both sides of the equation by (x - 2)(x + 4).]

A + B = 1, -2A + 4B = 22
[Compare the coefficients of like terms on both sides.]

(11-24) (AB) = (122) where C = (11-24), X = (AB), and D = (122)
[Write the system of equations in matrix form as CX = D.]

X = C-1D

(AB) = (11-24)-1 (122)
[Use matrix inversion method.]

= 1 / 6(4-121) (122)
[Use (abcd)-1 = 1ad - bc(d-b-ca).]

= (-34)

So, A = - 3, B = 4.


Correct answer : (1)
 3.  
Which of the following could be the graph of the function y = 2 + 1x + 5 - 2x - 4?

a.
Graph B
b.
Graph A


Solution:

y = 2 + 1x + 5 - 2x - 4

Clearly, at x = -5 and at x = 4, the given function y is not defined.

At x = 0, y = 2.7
[Put x = 0 to find y in the equation of the curve.]

From the given graphs, Graph A has the y-intercept of 2.7 approximately.

So, Graph A could be the graph of the given function.


Correct answer : (2)
 4.  
Which of the following graphs matches with the graph of the function y = -1 + 1x + 5 - 1x - 4?


a.
Graph A
b.
Graph B


Solution:

y = -1 + 1x + 5 - 1x - 4

Clearly, at x = -5 and at x = 4, the given function y is not defined.

At x = 0, y = - 0.55
[Put x = 0 to find y in the equation of the curve.]

From the given graphs, Graph B has the y - intercept of - 0.55 approximately.

So, Graph B could be the graph of the given function.


Correct answer : (2)
 5.  
Find the single reduced fraction of 2x(3x + 4)-3 (x - 3).
a.
2x2(3 x + 4)(x + 3 )
b.
2x2 - 15x - 12(3 x + 4)(x - 3 )
c.
15x(3 x + 4)
d.
15x - 12(3 x + 4)(x - 3 )


Solution:

2x(3x + 4)-3(x - 3) = 2x(x - 3) - 3(3x + 4)(3x + 4)(x - 3)

= 2x2 - 6x - 9x - 12(3x + 4)(x - 3)

= 2x2 - 15x - 12(3x + 4)(x - 3)


Correct answer : (2)
 6.  
Find the partial fraction decomposition of 5(x + 2)(x + 3).
a.
5(x + 2)-4(x + 3)
b.
5(x + 2)-5(x + 3)
c.
5(x + 2)+5(x + 3)
d.
None of the above


Solution:

5(x + 2)(x + 3) = A(x + 2)+B(x + 3)

5(x + 2)(x + 3) = A(x + 3) + B(x + 2)(x + 2)(x + 3)

5 = A(x + 3) + B(x + 2)

Then, A + B = 0; 3A + 2B = 5
[Compare the co-efficients of like terms on both sides.]

(1132) (AB) = (05)
    C         X    =    D
[Write system of equations in matrix form as C X = D.]

X = C - 1 · D

(AB) = (1132)-1 (05)
[Use matrix inversion method.]

= - 1(2-1-31) (05)
[Inverse of (abcd) = 1ad - bc(d-b-ca).]

= (5-5)

So, A = 5 and B = - 5.

So, 5(x + 2)(x + 3) = 5(x + 2)-5(x + 3)


Correct answer : (2)
 7.  
Find A + B + C, if f(x) = Ax2 + Bx + C + 5 and g(x) = 7x2 - 5x + 17 such that f(x) = g(x).
a.
- 14
b.
14
c.
- 7
d.
7


Solution:

f(x) = Ax2 + Bx + C + 5 and g(x) = 7x2 - 5x + 17

Ax2 + Bx + C + 5 = 7x2 - 5x + 17
[f(x) = g(x).]

A = 7; B = - 5; C + 5 = 17 C = 12
[Compare the coefficients of like terms on both sides.]

So, A + B + C = 7 - 5 + 12 = 14


Correct answer : (2)
 8.  
Choose the form of the partial fraction decomposition of x + 46(x + 8)(x - 4), from the following.
a.
Ax+Bx + 8 + Cx - 4 
b.
Ax + 8 + Bx - 4 
c.
Ax + 8 + Bx + 4 
d.
Ax - 8 + Bx - 4 


Solution:

x + 46(x + 8)(x - 4) = Ax + 8 + Bx - 4
[Decompose the fraction with distinct linear factors.]


Correct answer : (2)
 9.  
Choose the form of the partial fraction decomposition of 5x - 3x2(x + 5)(x2+6)from the following.
a.
Ax +Bx2 +Cx+5  + D
b.
Ax +Bx2 +Cx+5 +Dx + Ex2 + 6
c.
Does not exist
d.
Ax +Bx2 +Cx+5 +Dx2 + 6


Solution:

5x - 3x2(x + 5)(x2+6)

= Ax + Bx2 + Cx + 5 + Dx + Ex2+6
[Write the partial fraction decomposition. ]

So, 5x - 3x2(x + 5)(x2+6) = Ax + Bx2 + Cx + 5 + Dx + Ex2+6.


Correct answer : (2)
 10.  
Write the form of the partial fraction decomposition of x2 + 6x + 5(x - 2)3.
a.
Ax - 2  + B(x - 2)2 + C(x - 2 )3
b.
Does not exist
c.
Ax - 2  + B(x - 2)2 + C x + D(x - 2 )3
d.
C(x - 2 )3


Solution:

x2+6x + 5(x - 2)3

Ax - 2 + B(x - 2)2 + C(x - 2)3
[Write the partial fraction decomposition produced by the factor (x - 2)3.]

So, x2+6x + 5(x - 2)3 = Ax - 2 + B(x - 2)2 + C(x - 2)3.


Correct answer : (1)

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