# Adding And Subtracting Rational Expressions

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Rational expression is a fraction of two polynomials
When polynomials in the denominator ≠ 0.

For example: - (x^1 + 2)/ (x^2 + 1) is a rational expression where x^2 + 1 ≠ 0.
Other examples of rational expressions like

• (x + 1) / (x + 2)
• (3x^2 + 2) / (x + 5)
• (9x^3 + 6x^2 + 8x + 5) / (7x^2 + 5x + 9)
• 3/ x
• 5 / (x + 1)

Few examples those are not a rational expressions: -

[5 + x^(3/2)] / (x + 9): - It is not a rational expression because 5 + x^ (3/2) is not a polynomial. Polynomial defines that the power should be non- negative intger but 3/ 2 is not an integer.

[1 + x ^ (-1)] / x: - -1 is a negative number so numerator is not a polynomial. Hence it is not a rational expression.

2x / [x ^ (√2) + 5]: - √2 is not integer so it is not a rational expression.

Examples of Adding and subtracting rational expressions: -

Add the following rational expressions: -

(3x + 2) / x^2,  (x^2 + 2) / x^3.

Solution: -

(3x + 2) / x^2 + (x^2 + 2) / x^3    =[ x (3x + 2) + (x^2 + 2)] / x^3
= (3x^2 + 2x + x^2 + 2) / x^3
= (4x^2 +2x + 2) / x^3
= 2(2x^2 + x + 1) / x^3

·         Simplify: - (x+1)/(x-1) – (x-1)/(x+1)

Solution: - [(x+1)^2-(x-1)^2]/[(x+1)(x-1)]
=[(x^2+2x+1)-(x^2-2x+1)]/(x^2-1^2)
=(x^2+2x+1-x^2+2x-1)/(x^2-1)
=4x/(x^2-1)