Download Practice Test over Rational Expressions

Intermediate Algebra, MATH 0303
Practice Test over
Operations with Rational Expressions & Solving Equations with Rational Expressions
#1)
Reduce:
2x 2 + 7x + 3
x+3
#2)
Reduce:
6 xz + 15 x − 14 z − 35
6 x 2 + x − 35
#3)
Divide:
3x 2 + 7 x + 4
x+2
#4)
Divide:
a3 − 8
a−2
#5)
Combine:
#6)
Combine:
2
3
1
+ 2
+
x + 3 x + 7 x + 12 x + 4
5 4
−
3t t 2
y 2 + 2 y − 24 y 2 − y − 6 y 2 + 3 y − 18
⋅
÷
y 2 + 3 y + 2 y 2 + y − 20 y 2 + 6 y + 5
#7)
Perform the indicated operations:
#8)
Perform the indicated operations:
2 + y y 2 − 16
⋅
y − 4 8 + y3
#9)
Perform the indicated operations:
xy + 7 x − 3 y − 21 x 2 + 4 x + 3
⋅
xy + y + 7 x + 7
x 2 − 2x − 3
4−
#10)
Simplify the complex fraction:
4+
1
x2
4 1
+
x x2
#11)
Simplify the complex fraction:
x−
1
x−
1
2
#12)
Simplify the complex fraction:
1
1
+
x+a x−a
1
1
−
x+a x−a
#13)
Solve the equation:
5
2 1
= −
2 x x 12
#14)
Solve the equation:
1
2
− 2y
−
= 2
y +2 y −3 y − y −6
y−2
y −3
#15)
Solve the equation for y:
x=
#16)
Solve the equation for P:
1 1 1 1
= + +
P Q R S
SOLUTIONS
#1)
#2)
Reduce:
2 x 2 + 7 x + 3 (2 x + 1)( x + 3)
=
=
x+3
x+3
Reduce:
6 xz + 15 x − 14 z − 35
6 x 2 + x − 35
3x(2 z + 5) − 7(2 z + 5)
(2 x + 5)(3x − 7)
(3x − 7)(2 z + 5)
(2 x + 5)(3x − 7)
2z + 5
2x + 5
2x + 1
3x + 1
x + 2 3x + 7 x + 4
2
3x + 7 x + 4
→
x+2
2
#3)
Divide:
3x 2 + 6 x
x +4
x + 2
2
3x + 1 +
2
x+2
a 2 + 2a + 4
a − 2 a 3 + 0a 2 + 0a − 8
a 3 − 2a 2
a −8
Þ
a−2
3
#4)
Divide:
a 2 + 2a + 4
2a 2 + 0a
2a 2 − 4a
4a − 8
4a − 8
0
2
3
1
+ 2
+
x + 3 x + 7 x + 12 x + 4
2
3
1
+
+
x + 3 ( x + 3)( x + 4) x + 4
#5)
Combine:
2( x + 4)
3
1( x + 3)
+
+
( x + 3)( x + 4) ( x + 3)( x + 4) ( x + 3)( x + 4)
2x + 8 + 3 + x + 3
( x + 3)( x + 4)
3x + 14
( x + 3)( x + 4)
#6)
Combine:
5 4
−
3t t 2
t 5 4 3
⋅ − ⋅
t 3t t 2 3
5t 12
−
3t 2 3t 2
5t − 12
3t 2
y 2 + 2 y − 24 y 2 − y − 6 y 2 + 3 y − 18
⋅
÷
y 2 + 3 y + 2 y 2 + y − 20 y 2 + 6 y + 5
#7)
Perform the indicated operations:
( y + 6)( y − 4) ( y + 2)( y − 3) y 2 + 6 y + 5
⋅
⋅
( y + 1)( y + 2) ( y + 5)( y − 4) y 2 + 3 y − 18
( y + 6)( y − 3) ( y + 1)( y + 5)
⋅
( y + 1)( y + 5) ( y − 3)( y + 6)
1
#8)
Perform the indicated operations:
2 + y y 2 − 16
⋅
y − 4 8 + y3
2+ y
( y + 4)( y − 4)
⋅
y − 4 (2 + y )(4 − 2 y + y 2 )
(2 + y )( y + 4)( y − 4)
( y − 4)(2 + y )(4 − 2 y + y 2 )
y+4
4 − 2y + y2
#9)
Perform the indicated operations:
xy + 7 x − 3 y − 21 x 2 + 4 x + 3
⋅
xy + y + 7 x + 7
x 2 − 2x − 3
( x + 1)( x + 3)
x( y + 7) − 3( y + 7)
⋅
( x + 1)( x − 3)
y ( x + 1) + 7( x + 1)
( x − 3)( y + 7) ( x + 1)( x + 3)
⋅
( x + 1)( x − 3) ( y + 7)( x + 1)
x+3
x +1
x2
⋅ 1
4 1 x2
4+ + 2
x x
1
2
4x −1
4x 2 + 4x + 1
(2 x + 1)(2 x − 1)
(2 x + 1)(2 x + 1)
4−
#10)
Simplify the complex fraction:
1
x2
2x −1
2x + 1
To simplify, multiply
the complex fraction
by one expressed as
the LCD/LCD.
x−
1
1
2
1
x−
x−
#11)
Simplify the complex fraction:
2x 1
−
2 2
1
x−
2x − 1
2
2
x−
2x − 1
2x − 1 x
2
⋅ −
2x − 1 1 2x − 1
2x 2 − x
2
−
2x − 1 2x − 1
Add the
denominator.
Remember that 1
divided by a number is
equal to 1 times the
reciprocal.
2x 2 − x − 2
2x − 1
To simplify, multiply the
complex fraction by one
expressed as the LCD/LCD.
Watch out for the subtraction
sign, which must be distributed
across polynomials.
#12)
Simplify the complex fraction:
1
1
(x + a )(x − a )
+
x+a x−a ⋅
1
(
1
1
x + a )( x − a )
−
1
x+a x−a
x−a+x+a
x − a − ( x + a)
x+x−a+a
x−a−x−a
2x
x−x−a−a
2x
− 2a
−
x
a
To solve equations with rational
expressions, multiply the equation
by the LCD.
#13)
Solve the equation:
5
2 1
= −
2 x x 12
2 1ö
æ 5
= − ÷
12 xç
è 2 x x 12 ø
30 = 24 − x
6 = −x
−6 = x
x = −6
#14)
Solve the equation:
1
2
− 2y
−
= 2
y + 2 y −3 y − y −6
æ 1
ö
2
− 2y
çç
÷÷ ⋅ ( y + 2)( y − 3)
−
=
è y + 2 y − 3 ( y + 2)( y − 3) ø
y − 3 − 2( y + 2) = −2 y
y − 3 − 2 y − 4 = −2 y
y − 2 y − 3 − 4 = −2 y
− y − 7 = −2 y
− 7 = −2 y + y
− 7 = −y
7= y
y=7
x=
#15)
Solve the equation for y:
After multiplying the equation
by the LCD, put all the ys on
one side of the equation, factor
out a y and divide.
y−2
y −3
æ
y −2ö y −3
çç x =
÷⋅
y − 3 ÷ø 1
è
x( y − 3) = y − 2
xy − 3x = y − 2
xy − y = −2 + 3x
y ( x − 1) = −2 + 3x
y=
− 2 + 3x
x −1
1 1 1 1
= + +
P Q R S
#16)
Solve the equation for P:
æ 1 1 1 1 ö PQRS
çç = + + ÷÷ ⋅
1
èP Q R Sø
QRS = PRS + PQS + PQR
QRS = P( RS+QS+QR)
QRS
=P
RS+QS+QR
P=
QRS
RS+QS+QR