SIMPLIFYING RATIONAL EXPRESSIONS

An expression of the form f(x)/g(x) where f(x) and g(x) are two polynomials over the set of real numbers and g(x) ≠ 0 is called a rational expression.

Examples :

Simplify each of the following rational expressions :

Example 1 :

4x³/8x²

Solution :

= 4x³/8x²

Here 4 and 8 are divisible by 4, after cancelling the common factor. We get

= x/2

Example 2 :

(5x + 20) / (7x + 28)

Solution :

= (5x + 20) / (7x + 28)

In the numerator, the common factor is 5. At the denominator the common factor is 7. After factoring common factors, we get

= 5(x + 4) / 7(x + 4)

= 5/7

So, the answer is 5/7

Example 3 :

(3x + 9) / (3x + 15)

Solution :

= (3x + 9) / (3x + 15)

Common factor for the numerator is 3 and the common factor for the denominator is also 3.

Factoring it out, we get

= 3(x + 3) / 3(x + 4)

= (x + 3)/(x + 4)

Example 4 :

(x - 3)/(x² - 9)

Solution :

= (x - 3)/(x² - 9)

The denominator looks like an algebraic identity a² - b²

a² - b² = (a + b)(a - b)

(x² - 9) = (x² - 3²)

= (x + 3)(x - 3)

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

= 1/(x + 3)

Example 5 :

Solution :

= (x² - 2x + 1)/(x - 1)

(x² - 2x + 1) = (x - 1)(x - 1)

= (x - 1)(x - 1)/(x - 1)

= (x - 1)

Example 6 :

Solution :

= (x² + 7x + 10)/(x² - 4)

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

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

Example 7 :

Solution :

= (9x² - 25y²)/(3x² - 5xy)

= ((3x)² - (5y)²)/x(3x - 5y)

= (3x + 5y)(3x - 5y)/x(3x - 5y)

= (3x + 5y)/x

Example 8 :

Solution :

= (6x² - 54) / (x² + 7x + 12)

6x² - 54 = 6(x² - 9)

= 6(x² - 3²)

= 6(x + 3)(x - 3) -----(1)

(x² + 7x + 12) = (x + 4)(x + 3) -----(2)

(1) / (2)

= 6(x + 3)(x - 3) / (x + 4)(x + 3)

= 6(x - 3)/(x + 4)

Example 9 :

Solution :

= (2 - x) / (x² + 4x - 12)

x² + 4x - 12 = x² + 6x - 2x - 12

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

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

From the numerator (2 - x), we factor negative.

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

(1) / (2)

= -(x - 2) / (x - 2)(x + 6)

= -1/(x - 6)

Example 10 :

Solution :

= (64a³ + 125b³) / (4a² b + 5ab²)

Simplifying the numerator :

64a³ + 125b³ = (4a)³ + (5b)³

a³ + b³ = (a + b)(a² - ab + b²)

(4a)³ + (5b)³ = (4a + 5b) ((4a)² + (4a)(5b) + (5b)²)

= (4a + 5b) (16a² + 20ab + 25b²) -----(1)

Simplifying the denominator :

4a² b + 5ab² = ab(4a + 5b) -------(2)

(1) / (2)

= (4a + 5b) (16a² + 20ab + 25b²) / ab(4a + 5b)

= (16a² + 20ab + 25b²) / ab

Example 11 :

Solution :

= (x⁴ - 16)/(x² + 5x + 6)

(x²)² - 4² = (x² + 4)(x² - 4)

= (x² + 4)(x² - 2²)

= (x² + 4)(x + 2)(x - 2) ------(1)

x² + 5x + 6 = x² + 2x + 3x + 6

= x(x + 2) + 3(x + 2)

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

(1) / (2)

= (x² + 4)(x + 2)(x - 2) / (x + 2)(x + 3)

= (x² + 4)(x - 2) / (x + 3)

Example 12 :

Solution :

= (v³ + 11v² + 18v)/(v² + v - 2)

Simplifying the numerator :

= v(v² + 11 v + 18)

= v(v² + 9v + 2v + 18)

= v[v(v + 9) + 2(v + 9)]

= v(v + 2)(v + 9) -------(1)

Simplifying the denominator :

v² + v - 2 = v² + 2v - v - 2

= v(v + 2) - 1(v + 2)

= (v - 1)(v + 2) ------(2)

(1) / (2)

= v(v + 2)(v + 9)/(v - 1)(v + 2)

= v(v + 9) / (v - 1)

Example 13 :

Solution :

Example 14 :

Solution :

Example 15 :

Solution :

Example 16 :

Solution :

Example 17 :

Solution :

Example 18 :

Solution :

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