FC020 Week 3 Tutorial Questions

C3 1

Simplify a

e 2

c e

g

4

b

a2 − a −6

6 x + 15 8 x + 20

f

a 2 + 3a − 18

c

x 2 + 3x

g

2x 2 + 5x − 3

3y + 3

d

y2 + 7y + 6 3t 2 − 11t − 4

h

t 2 − 16

x 2 − 25 x 2 − 7 x + 10 6 x 2 − 13 x + 6 12 x 2 + x − 6

3x 2 4x − 4 × 9x − 9 2x n 2 + 2n n 2 + 6n + 8

4y 2 2y2 + y

÷

b

n+4

×

d

n2

y 2 + 2 y − 15

f

2 y 2 + 11 y + 5

4 − 3x − x 2 10 x − 10 ÷ 2 5 x + 15 x + 7 x + 12

h

x 2 − 36

x−6 x+2

÷

2

x + 7 x + 10 4 x − 12 x2 − 4

×

x 2 + 2x x 2 − 2x − 3

x 2 −1 2x 2 + 7 x − 4 a 3 − 3a 2 8a 2 − 4 a

6 x 2 − 5x + 1

×

3x 2 − 4 x + 1 a2 −9

÷

2 a 2 + 5a − 3

Express as a single fraction in its simplest form a

2 7 + y y+4

b

d

x 5 + ( x − 3)( x − 1) 2( x − 1)

e

g

4 x + 2 x+5 x + 8 x + 15

h

2x 1 – x−5 x+3 2

+

2

q + 3q 6x 2

x −4

–

5q 4q + 12

3 x+2

c

7 3x – x( x + 2) x+2

f

4 x+2 + 2 3x − 3 x −x 5t + 12

i

2

2t + 7t + 3

–

4 2t + 1

Simplify a e

5

3x − 1 18 x − 6

Express as simply as possible a

3

Worksheet A

RATIONAL EXPRESSIONS

x 2 − 5x 6 x − 30

b

3x − x 2

f

2 x 2 − 18

16 − x 2 x 2 + 2x − 8 x 3 + x 2 − 2x 3x 2 + 4 x − 4

c g

2x 2 − 4x − 6

d

3 x 2 − 12 x + 9 2 + 5 x − 3x 2

h

2 x 2 + x − 10

x3 − x 2x 2 − x −1 x 4 − 5x 2 + 4 x2 − x − 2

Express as simply as possible a

c

e

10 x 2 − 10 x 2 + 6x + 8 × 2 5 x + 10 x + 5x + 4 2 x 2 + 12 x + 10 4x 2 − 7x + 3 x 4 + 6x 2 + 5 x2 −9

×

÷

4 x 2 + 20 x 4 x 2 − 3x

2x 2 − 6x 4x 2 + 4

b

d

f

t 2 − 2t 2t 2 − t − 6

÷

8x 2 + 6x − 9 4 x 2 + 12 x + 9 y 4 − 16 5y 2 + 9y − 2

 Solomon Press

9t 2 − 4 6t 2 + 13t + 6

×

÷

2 x 2 + 3x 6 − 8x y2 + 4 25 y 2 − 10 y + 1

C3

RATIONAL EXPRESSIONS

6

Express as a single fraction in its simplest form a c e g

7

5 2

x −1

–

Worksheet A continued

1 2x + 2

4 2

x + 2x − 3

b 1

+

d

2

x − 3x + 2

2x − 1 x + 3x + 6 x + 4x + 4

f

2

x 2

x − x−6

2

+

h

2

2 x + 3x − 2

3x

–

2

x −4 x +1 2

x − 25

4 2

2 x + 3x − 2 2

+

2

x + 5x

3 x 1 + 2 + 2 x − 6x + 9 x−3 x − 3x 1 x

2

1

–

+

2

3x − 2 x

3 6x − 4

Solve 3 2 = x 2x − 5

a 1– c

20

10 2x + 1

+1=

2

2x + 5x + 2

e 5+

b

1 2

x + 5x + 6

=

11 x+3 7 x − 15

f(x) ≡

8

Show that f(x) =

2

x − 5x

f

–

4 , x−5

x −1

+

3 =1 x +1

y +3 1 2 y −1 – = y+5 2 y 3 1 − 4x + 4x

2

–

10 1− 4x

2

=

0 < x < 5.

k , where k is an integer to be found. x

f(x) ≡

9

d

2 2

x−5 2

3x + 5 x − 2

+

2 , 3x − 1

x > 1.

Show that f(x) =

1 . x+2

10

Given that f(x) ≡

8x x+2 1 = 2 , x ≠ ± 2, show that f(x) – . x−2 f ( x) x −4

11

a Express

2 3 + as a single fraction in its simplest form. x+5 ( x + 2)( x + 5)

b Hence solve the equation 2 3 1 = , + x+5 ( x + 2)( x + 5) 3

giving your answers to 2 decimal places. 5 3 – = 2 has no real roots. 2 4x + 2 4x + 4x + 1

12

Show that the equation

13

1Ô x −1 Ã 6 – Õ ÷ 2 as a single fraction in its simplest form. Express Ä x − 25 Å x+5 xÖ

 Solomon Press

5 1 + 2x

C3 1

2

Worksheet B

RATIONAL EXPRESSIONS Find the quotient and remainder obtained in dividing a (3x3 − 10x2 − 9x + 15) by (x − 4)

b (2x3 − 11x2 − x + 3) by (2x − 1)

c (4x3 + 8x2 + 7x + 32) by (2x + 5)

d (1 − 22x2 − 6x3) by (3x + 2)

a Show that (x + 2) is a factor of (x3 + 4x2 + x − 6). b Fully factorise x3 + 4x2 + x − 6. c Simplify

3

a Show that (2x − 3) is a factor of (2x3 − 5x2 + 13x − 15). b Simplify

4

2 x3 − 5 x 2 + 13 x − 15 . 2 x2 − 7 x + 6

x3 − 1 . x2 + x − 2

Find the integers A and B such that 2x + 5 x+3

6

8

A+

B . x+3

Express each of the following in the form A + a

7

≡

a State a linear factor of x3 − 1. b Simplify

5

x3 + 4 x 2 + x − 6 . x2 − 9

x+2 x +1

b

x+3 x−2

c

, where f(x) is linear.

d

2x + 1 x+2

e

x −1 2x − 1

f

1 − 4x 3 + 2x

Find the quotient and remainder obtained in dividing a (x2 + 3x + 5) by (x2 + x + 2)

b (2x2 + 3x − 8) by (x2 − x − 4)

c (x2 + 7) by (x2 + 3x − 1)

d (3x2 − x − 4) by (x2 + 2)

≡≠

e (x3 − 2x2 − 5x + 8) by (x2 + x − 2)

f (2x3 − 7x2 + 1) by (x2 − 5x + 1)

g (3x3 + 6x2 − 2x + 5) by (3x2 + 4)

h (6x3 − x2 − 44x − 6) by (2x2 − 5x − 2)

a Divide (x3 + 5x2 + 7x − 13) by (x2 + 3x − 4). b Hence show that x3 + 5 x 2 + 7 x − 13 x 2 + 3x − 4

9

x 1− x

B f ( x)

f(x) =

x+2+

5 . x+4

x3 − 2 x 2 − 21x + 70 , x x 2 + 2 x − 15

a Express f(x) in the form Ax + B +

C g( x)

3.

, where g(x) is linear.

b Hence, or otherwise, solve the equation f(x) =

3x − 7 . x−3

 Solomon Press