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 the following rational expressions :

Example 1 :

4x³/8x²

Solution :

= 4x³/8x²

= (4/8)x^3-2

= (1/2)x

= x/2

Example 2 :

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

Solution :

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

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

= 5/7

Example 3 :

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

Solution :

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

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

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

Example 4 :

(x - 3)/(x² - 9)

Solution :

= (x - 3)/(x² - 9)

= (x - 3)/(x² - 3²)

= (x - 3)/[(x + 3)(x - 3)]

= 1/(x + 3)

Example 5 :

(x² - 2x(1) + 1²)/(x - 1)

Solution :

= (x² - 2x(1) + 1²)/(x - 1)

= (x - 1)²/(x - 1)

= (x - 1)

Example 6 :

(x² + 7x + 10)/(x² -  4)

Solution :

= (x² + 7x + 10)/(x² -  4)

= [(x + 5)(x + 2)]/(x² - 2²)

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

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

Example 7 :

(9x² - 25y²)/(3x² - 5xy)

Solution :

= (9x² - 25y²)/(3x² - 5xy)

= [(3x)² - (5y)²]/x(3x - 5y)

= [(3x + 5y)(3x - 5y)]/x(3x - 5y)

= (3x + 5y)/x

Example 8 :

(6x² - 54)/(x² + 7x + 12)

Solution :

= (6x² - 54)/(x² + 7x + 12)

= 6(x² - 9)/[(x + 3)(x + 4)]

= 6(x² - 3²)/[(x + 3)(x + 4)]

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

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

Example 9 :

(2 - x)/(x² + 4x - 12)

Solution :

= (2 - x)/(x² + 4x - 12)

= (2 - x)/[(x + 6)(x - 2)]

= -(x - 2)/[(x + 6)(x - 2)]

= -1/(x + 6)

Example 10 :

(64a³ + 125b³)/(4a²b + 5ab²)

Solution :

= (64a³ + 125b³)/(4a²b + 5ab²)

= [(4a)³ + (5b)³]/[ab(4a + 5b)]

= {(4a + 5b)[(4a)² - (4a)(5b) + (5b)²]}/[ab(4a + 5b)]

= (16a² - 20ab + 25b²)/ab

Example 11 :

(x⁴ - 16)/(x² + 5x + 6)

Solution :

= (x⁴ - 16)/(x² + 5x + 6)

=  [(x²)² - 4²]/[(x + 2)(x + 3)]

= [(x² + 4)(x² - 4)]/[(x + 2)(x + 3)]

= [(x² + 4)(x² - 2²)]/[(x + 2)(x + 3)]

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

= [(x² + 4)(x - 2)]/(x + 3)

Example 12 :

(v³ + 11v² + 18v)/(v² + v - 2)

Solution :

= (v³ + 11v² + 18v)/(v² + v - 2)

= [v(v² + 11v + 18)]/(v² + v - 2)

= [v(v + 9)(v + 2)]/[(v + 2)(v - 1)]

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

Example 13 :

(2x³ + 16x² + 24x)/(x² - x - 6)

Solution :

= (2x³ + 16x² + 24x)/(x² - x - 6)

= [2x(x² + 8x + 12)]/(x² - x - 6)

= [2x(x + 6)(x + 2)]/[(x - 3)(x + 2)]

= 2x(x + 6)/(x - 3)

Example 14 :

(xy + 3x - 2y - 6)/(y² + y - 6)

Solution :

= (xy + 3x - 2y - 6)/(y² + y - 6)

= [x(y + 3) - 2(y + 3)]/[(y + 3)(y - 2)]

= [(y + 3)(x - 2)]/[(y + 3)(y - 2)]

= (x - 2)/(y - 2)

Example 15 :

(ax - ay + bx - by)/(ax - ay - bx + by)

Solution :

= (ax - ay + bx - by)/(ax - ay - bx + by)

= [a(x - y) + b(x - y)]/[a(x - y) - b(x - b)]

= [(x - y)(a + b)]/[(x - y)(a - b)]

= (a + b)/(a - b)

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