CBSE Class 10 Mathematics Rational Expressions Notes Set 15

Rational Expressions

A rational expression is a fraction \( \dfrac{f(x)}{g(x)} \) in which both f(x) and g(x) are polynomials and g(x) is not the zero polynomial.
Every polynomial is a rational expression because f(x) = f(x)/1. If the numerator or the denominator is not a polynomial (for example \( \dfrac{\sqrt x}{x+1} \)), the fraction is not a rational expression.

Lowest Terms

A rational expression is in lowest terms when the numerator and the denominator have no common factor other than 1. To reduce it, factorise both parts completely and cancel the common factors (never common terms).
Example: \( \dfrac{x^2+8x+12}{x^2+4x-12}=\dfrac{(x+2)(x+6)}{(x+6)(x-2)}=\dfrac{x+2}{x-2} \).

Operations

1. Addition: \( \dfrac{f}{g}+\dfrac{h}{p}=\dfrac{fp+hg}{gp} \); with equal denominators simply add the numerators.
2. Subtraction: \( \dfrac{f}{g}-\dfrac{h}{p}=\dfrac{fp-hg}{gp} \).
3. Multiplication: \( \dfrac{f}{g}\times\dfrac{h}{p}=\dfrac{fh}{gp} \).
4. Division: multiply by the reciprocal of the divisor, \( \dfrac{f}{g}\div\dfrac{h}{p}=\dfrac{fp}{gh} \), with h not zero.
5. Sums, differences, products and quotients (by a non-zero expression) of rational expressions are again rational expressions.
6. The additive inverse of \( \dfrac{f}{g} \) is \( \dfrac{-f}{g} \).

Useful Factorisations

\( a^2-b^2=(a-b)(a+b) \),  \( a^3-b^3=(a-b)(a^2+ab+b^2) \),  \( a^3+b^3=(a+b)(a^2-ab+b^2) \),  \( (a+b)^2-(a-b)^2=4ab \).
To factorise \( x^2+px+q \) find two numbers with sum p and product q.

Worked Examples

Illustration 1. Simplify \( \dfrac{x+1}{2x-1}+\dfrac{3x-2}{x-1} \).
Answer:
Common denominator (2x − 1)(x − 1): numerator (x + 1)(x − 1) + (3x − 2)(2x − 1) = x² − 1 + 6x² − 7x + 2 = 7x² − 7x + 1.
Answer: \( \dfrac{7x^2-7x+1}{2x^2-3x+1} \).

Illustration 2. Reduce \( \dfrac{x^2-5x+6}{x^2-9x+20} \) to lowest terms.
Answer:
Factorise: \( \dfrac{(x-2)(x-3)}{(x-4)(x-5)} \). There is no common factor, so it is already in lowest terms.

Illustration 3. Simplify \( \dfrac{x^2-y^2}{x^2+2xy+y^2}\times\dfrac{xy+y^2}{x^2-xy} \).
Answer:
\( \dfrac{(x-y)(x+y)}{(x+y)^2}\times\dfrac{y(x+y)}{x(x-y)} \) = \( \dfrac{y}{x} \).