PROBLEMS ON SIMPLIFYING RATIONAL EXPRESSIONS

Simplify the rational expressions given below.

Problem 1 :

Solution :

Numerator :

9 - y² = 3² - y²

= (3 + y)(3 - y)

Denominator :

y² + 2y - 15 = y² + 5y - 3y - 15

= y(y + 5) - 3(y + 5)

= (y + 5)(y - 3)

Numerator / denominator = (3 + y)(3 - y) / (y + 5)(y - 3)

= -(3 + y)(y - 3) / (y + 5) (y - 3)

= - (3 + y) / (y + 5)

Problem 2 :

Solution :

Numerator :

3x³ + 18x² - 21x

Factoring 3x from all, we get

= 3x (x² + 6x - 7)

= 3x (x + 7)(x - 1)

Denominator :

x³ + 5x² - 14x 

Factoring x from all terms, we get

= x(x² + 5x - 14)

= x(x² + 7x - 2x - 14)

= x [x (x + 7) - 2(x + 7)]

= x(x - 2) (x + 7)

Numerator / denominator = 3x (x + 7)(x - 1) / x(x - 2) (x + 7)

= 3(x - 1) / (x - 2)

Problem 3 :

Solution :

Numerator :

2x² - 18x

Factoring 2x, we get

= 2x (x - 9)

Denominator :

4x³ - 32x² - 36x

Factoring 4x, we get

= 4x(x² - 8x - 9)

= 4x(x - 9)(x + 1)

Numerator / denominator = 2x (x - 9) / 4x(x - 9)(x + 1)

= 1 / 2(x + 1)

Problem 4 :

Solution :

Numerator :

5x² - 11x + 2

= 5x² - 10x - 1x + 2

= 5x (x - 2) - 1(x - 2)

= (5x - 1)(x - 2)

Denominator :

5x² - 7x - 6

= 5x² - 10x + 3x - 6

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

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

Numerator / denominator = (5x - 1)(x - 2) / (5x + 3) (x - 2)

= (5x - 1)(5x + 3)

Problem 5 :

Solution :

Numerator :

2x² - 7x + 3

= 2x² - 6x - 1x + 3

= 2x (x - 3) - 1(x - 3)

= (2x - 1)(x - 3)

Denominator :

2x² + 9x - 5

= 2x² + 10x - 1x - 5

= 2x(x + 5) - 1(x + 5)

= (2x - 1) (x + 5)

Numerator / denominator =  (2x - 1)(x - 3) /  (2x - 1) (x + 5)

= (x - 3) / (x + 5)

Problem 6 :

Solution :

Numerator :

x² + 2x - 24

= x² + 6x - 4x - 24

= x(x + 6) - 4(x + 6)

= (x + 6)(x - 4)

Denominator :

12 - 4x - x²

= -(x²+ 4x - 12)

= - (x + 6)(x - 2)

Numerator / Denominator = (x + 6)(x - 4) / [- (x + 6)(x - 2)]

= - (x - 4) / (x - 2)

Problem 7 :

Solution :

Numerator :

2x² - 8

Factoring 2,

= 2(x² - 4)

= 2(x² - 2²)

= 2(x + 2) (x - 2)

Denominator :

x² + 4x - 12

= x² + 6x - 2x - 12

= x(x + 6) - 2(x + 6)

= (x - 2)(x + 6)

Numerator / denominator = 2(x + 2) (x - 2) / (x - 2)(x + 6)

= 2 (x + 2) / (x + 6)

Problem 8 :

Solution :

Numerator :

2x² - 9x + 7

= 2x² - 2x - 7x + 7

= 2x (x - 1) - 7(x - 1)

= (2x - 7)(x - 1)

Denominator :

x² + 2x - 3

= x² + 3x - 1x - 3

= x(x + 3) - 1(x + 3)

= (x - 1) (x + 3)

Numerator / denominator = (2x - 7)(x - 1) / (x - 1) (x + 3)

= (2x - 7) / (x + 3)

Problem 9 :

Solution :

Numerator :

8x² - 14x + 6

= 8x² - 8x - 6x + 6

= 8x

Factor 8x from the first two terms and factoring -6 from last two terms.

= 8x(x - 1) - 6(x - 1)

= (x - 1) (8x - 6)

= 2x(4x - 3) (x - 1)

Denominator :

(4x - 3)

Numerator / denominator = 2x(4x - 3) (x - 1) / (4x - 3)

= 2x (x - 1)