# How To Solve Rational Expressions

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Definition: - Rational expression is the ratio of two polynomial expressions.

Examples of rational expression: -

(x^3 + 3 x^2 + 3 x + 1) / (x ^2 + 2 x + 1)
(x ^2 – 2 x) / (x ^2 – 4)
x ^ 4 / 3 x ^2

Question 1: - Simplify x ^ 4 / 3 x ^2

Solution: - Cancel out the common factor of the numerator and denominator.
Common factor of x ^4 and 3 x^2 is x^2. Therefore
x ^ 4 / 3 x ^2 = x ^2 * x ^2 / 3 * x ^2 = x^2 / 3
Therefore x ^ 4 / 3 x ^2 = x^2 / 3

Question 2: - Simplify (x ^2 – 2 x) / (x ^2 – 4)

Solution: - Factorize the numerator and denominator.
i)             X ^2 – 2 x = x ( x – 2)
ii)            X ^ 2 – 4 = x ^2 – 2 ^2 = (x + 2) (x – 2)
Therefore, (x ^2 – 2 x) / (x ^2 – 4) = x ( x – 2) / (x + 2) (x – 2) = x / x + 2

Question 3: - Simplify (x^3 + 3 x^2 + 3 x + 1) / (x ^2 + 2 x + 1)

Solution: -
(x^3 + 3 x^2 + 3 x + 1) / (x ^2 + 2 x + 1)
= (x ^3 + 3 * x^2* 1 + 3 * x * 1^2 + 1 ^3) / (x ^2 + 2 * x * 1 + 1 ^2)
= (x + 1) ^3 / (x +1)^2
= (x+1)