# Partial Fractions Worksheet

Partial Fractions Worksheet
• Page 1
1.
Find the single reduced fraction of ( + ).
 a. b. c. d.

#### Solution:

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

= 9x - 22x2 - 5x + 6

2.
Choose the values of A & B such that = + .
 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.

3.
Which of the following could be the graph of the function $y$ = 2 + - ?

 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.

4.
Which of the following graphs matches with the graph of the function $y$ = -1 + - ?

 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.

5.
Find the single reduced fraction of .
 a. b. c. d.

#### 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)

6.
Find the partial fraction decomposition of .
 a. b. c. 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)

7.
Find A + B + C, if $f$($x$) = A$x$2 + B$x$ + C + 5 and $g$($x$) = 7$x$2 - 5$x$ + 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

8.
Choose the form of the partial fraction decomposition of , from the following.
 a. + b. + c. + d. +

#### Solution:

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

9.
Choose the form of the partial fraction decomposition of from the following.
 a. + D b. c. Does not exist d.

#### 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.

10.
Write the form of the partial fraction decomposition of .
 a. + + b. Does not exist c. + + d.

#### 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.