OPERATIONS WITH RATIONAL EXPRESSIONS WORKSHEET

Perform the operation and leave the answer in simplest form.

Simplify each expression.

Problem 1 :

3x + 2x2 - 9 · x + 33x2 + 5x + 2

Problem 2 :

2x2 + x - 62x2 + 9x + 9 ÷ 4x2 - 9x2 - 9

Problem 3 :

2x + 4 - 3x + 2

Problem 4 :

x + 3x - 2 + 2xx2 - 4

Problem 5 :

3x2- x - 6 - 2x2 + 5x + 6

Problem 6 :

3x + 7x2 - 16 - 2x + 3x2 - 16

Problem 7 :

4x - 35x + 2

Problem 8 :

2x - 18x + 2

Problem 9 :

6aa + 3 + 2a4

Problem 10 :

2m + 1 + 3m - 4

Solutions

1. Answer 

Given, 3x + 2x2 - 9 · x + 33x2 + 5x + 2= 3x + 2x2 - 32 · x + 33x2 + 3x + 2x + 2= 3x + 2(x + 3)(x - 3) · x + 33x(x + 1) + 2(x + 1)= 3x + 2(x + 3)(x - 3) · x + 3(3x + 2) (x + 1)= (3x + 2)(x + 3)(x + 3) (x - 3) (3x + 2) (x + 1)Cancel the common factor.= 3x + 2x + 3x + 3 (x - 3) 3x + 2 (x + 1)= 1(x - 3) (x + 1)= 1x2 + x - 3x - 3= 1x2 - 2x - 3

2. Answer 

Given, 2x2 + x - 62x2 + 9x + 9 ÷ 4x2 - 9x2 - 9= 2x2 + x - 62x2 + 9x + 94x2 - 9x2 - 9= 2x2 + x - 62x2 + 9x + 9 × x2 - 94x2 - 9= 2x2 + 4x - 3x - 62x2 + 6x + 3x + 9 × x2 - 324x2 - 9= 2x(x + 2) - 3(x + 2 )2x(x + 3) + 3(x + 3) × (x + 3) (x - 3)4x2 - 94= (2x - 3) (x + 2 )(2x + 3) (x + 3) × (x + 3) (x - 3)4x2 - 322= (2x - 3) (x + 2 )(2x + 3) (x + 3) × (x + 3) (x - 3)4x + 32x - 32= (2x - 3) (x + 2 ) (x + 3) (x - 3)(2x + 3) (x + 3) · 42x + 32 2x - 32= (2x - 3) (x + 2 ) (x + 3) (x - 3)(2x + 3) (x + 3) · 4(2x + 3) (2x - 3) 4= (2x - 3) (x + 2 ) (x + 3) (x - 3)(2x + 3) (x + 3) (2x + 3) (2x - 3)Cancel the common factor.= 2x - 3 (x + 2) x + 3 (x - 3)(2x + 3) x + 3 (2x + 3) 2x - 3= (x + 2)(x - 3)(2x + 3)2= x2 - 3x + 2x - 6(2x)2 + 32 + 2(2x)(3)= x2 - x - 64x2 + 12x + 9

3. Answer 

Given, 2x + 4 - 3x + 2= 2x + 4 × x + 2x + 2 - 3x + 2 × x + 4x + 4= 2(x + 2)(x + 2)(x + 4) - 3(x + 4)(x + 2)(x + 4)= 2(x + 2) - 3(x + 4)(x + 2)(x + 4)= 2x + 4 - 3x - 12x2 + 4x + 2x + 8= -x - 8x2 + 6x + 8

4. Answer 

Given, x + 3x - 2 + 2xx2 - 4= x + 3x - 2 + 2xx2 - 22= x + 3x - 2 + 2x(x + 2)(x - 2)= x + 3x - 2 × x + 2x + 2+ 2x(x + 2)(x - 2)= (x + 3)(x + 2)(x - 2)(x + 2) + 2x(x + 2)(x - 2)= (x + 3)(x + 2) + 2x(x + 2)(x - 2)= x2 + 2x + 3x + 6 + 2x (x + 2)(x - 2)= x2 + 5x + 6 + 2x(x + 2)(x - 2)= x2 + 7x + 6(x + 2)(x - 2)= x2 + 7x + 6x2 - 4

5. Answer 

Given, 3x2- x - 6 - 2x2 + 5x + 6= 3x2 + 2x - 3x - 6 - 2x2 + 3x + 2x + 6= 3x(x + 2) - 3(x + 2) - 2x(x + 3) + 2(x + 3)= 3(x - 3)(x + 2) - 2(x + 2)(x + 3)= 3(x - 3)(x + 2) × x + 3x + 3 - 2(x + 2)(x + 3) × x - 3x - 3= 3(x + 3)(x - 3)(x + 2)(x + 3) - 2(x - 3)(x + 2)(x + 3)(x - 3) = 3(x + 3) - 2(x - 3)(x - 3)(x + 2)(x + 3) = 3x + 9 - 2x + 6(x - 3)(x + 2)(x + 3) = x + 15(x - 3)(x + 2)(x + 3) = x + 15x2 + 2x - 3x - 6(x + 3) = x + 15x2 - x - 6(x + 3) = x + 15x3 + 3x2 - x2 - 3x - 6x - 18 = x + 15x3 + 2x2 - 9x - 18

6. Answer 

Given, 3x + 7x2 - 16 - 2x + 3x2 - 16= (3x + 7) - (2x + 3)x2 - 16= 3x + 7 - 2x - 3x2 - 16= x + 4x2 - 42= x + 4(x + 4)(x - 4)Cancel the common factor.= x + 4 (x + 4)(x - 4)= 1x - 4

7. Answer 

Given, 4x - 35x + 2= 4x - 3 × x + 25= 4(x + 2)5(x - 3)= 4x + 85x - 15

8. Answer 

Given, 2x - 18x + 2= 2x - 18 + 2xx = 2x - 1 × x8 + 2x= 2x(x - 1)(8 + 2x)= 2x8x + 2x2 - 8 - 2x= 2x2x2 + 6x - 8= 2x2x2 + 3x - 4= xx2 + 3x - 4

9. Answer 

Given, 6aa + 3 + 2a4= 6aa + 3 + a2= 6aa + 3 × 22+ a2 × a + 3a + 3= 12a2(a + 3) + a(a + 3)2(a + 3)= 12a + a(a + 3)2(a + 3) = 12a + a2 + 3a2a + 6 = 15a + a22a + 6

10. Answer 

Given, 2m + 1 + 3m - 4= 2m + 1 × m - 4m - 4 + 3m - 4 × m + 1m + 1= 2(m - 4)(m + 1)(m - 4) + 3(m + 1)(m - 4)(m + 1)= 2(m - 4) + 3(m + 1)(m + 1)(m - 4)= 2m - 8 + 3m + 3m2 - 4m + m- 4= 5m - 5m2 - 3m - 4

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