How to Simplify Rational Expressions

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Rational expression is a ratio of two polynomial expressions.
 
For example: -
·3 x^2 / 6x
·(x^2 + 4 x + 4) / (2 x + 4)
·(x^2 – 9) / (x^3 -9 x^2 +27 x – 9) and so on.
 
How to simplify rational expressions: -


Example 1: - Simplify 3 x^2 / 6x
Solution: -
Cancel out the common factor of the numerator and denominator.
Common Factor of 3x^2 and 6x is 3x.
Hence we can write
3 x^2 / 6x = 3x *x / 3x * 2 = x /2
Answer: - Therefore, 3 x^2 / 6x = x / 2.


Example 2: - Simplify (X^2 + 4 x + 4) / (2 x + 4)
 
Solution: -
i) Factor the numerator.
Therefore (X^2 + 4 x + 4) = x^2 + 2 * x^2 * 2 + 2^2 = (x + 2)^2
[Since (a + b)^2 = a^2 + 2ab + b^2]
 
ii) Take the common factor of the denominator
Therefore (2 x + 4) = 2 (x + 2).
 
iii) Cancel out the common term of the numerator and denominator, therefore we can write,
(X^2 + 4 x + 4) / (2 x + 4) = (x + 2)^2 / 2 (x + 2) = (x + 2) / 2

Example 3: - Simplify (x^2 – 9) / (x^3 -9 x^2 +27 x – 9)
Solution: - Similarly
(x^2 – 9) / (x^3 -9 x^2 +27 x – 9) = (x+3) (x - 3)/(x - 3) ^3 = (x+3)/(x-3) ^2

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