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 :
The following steps will be useful to simplify rational expressions.
Step 1 :
Factor both numerator and denominator, if possible.
Step 2 :
Identify the common factors in both numerator and denominator.
Step 3 :
Remove the common factors found in both numerator and denominator.
Simplify the following rational expressions :
Example 1 :
4x3/8x2
Solution :
= 4x3/8x2
= (4/8)x3-2
= (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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