What are algebraic fractions?

In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. For example, $$\frac{4x}{x^{2}+3x-4}$$ is an algebraic fraction. Algebraic fractions are subject to the same law as arithmetic fractions.

An algebraic fraction whose numerator and denominator are both polynomials is called a rational fraction. Thus, $$\frac{4x}{x^{2}+3x-4}$$ is a rational fraction, but not $$\frac{\sqrt{x+2}}{x^{2}+3x-4}$$ because the numerator contains a square root function.

E2.3A: Manipulate algebraic fractions

In the algebraic fraction $$ \frac{a}{b}$$, the dividend $$a$$ is called the numerator, and the divisor $$b$$ is called the denominator.

The standard form of the rational expression is $$\frac{A\left ( x \right )}{B\left ( x \right )}$$, where $$ A\left ( x \right )$$ and $$B\left ( x \right )$$ are the polynomial expressions and $$B\left ( x \right )\neq 0$$.

Worked examples of algebraic fractions

Example 1: Evaluate $$\frac{x}{3}+\frac{x-4}{2}$$.

Step 1: Write the given algebraic fractions.

$$\frac{x}{3} +\frac{x-4}{2}$$

Step 2: Make denominators of both the fractions equal.

$$\frac{x}{3} +\frac{x-4}{2}=\frac{2x}{6} +\frac{3x-12}{6}$$

Step 3: Solve the newly formed expression.

$$\frac{2x}{6} +\frac{3x-12}{6}=\frac{2x+3x-12}{6}$$

$$\frac{2x}{6}+\frac{3x-12}{6}=\frac{5x-12}{6}$$

Step 4: Write the final answer.

$$\frac{2x}{6}+\frac{3x-12}{6}=\frac{5x-12}{6}$$

Example 2: Evaluate $$\frac{2x}{3}-\frac{3\left(x-5 \right )}{2}$$.

Step 1: Write the given algebraic fractions.

$$\frac{2x}{3} -\frac{3\left(x-5 \right )}{2}$$

Step 2: Make denominators of both the fractions equal.

$$\frac{2x}{3}-\frac{3\left(x-5 \right )}{2}=\frac{4x}{6} -\frac{9\left(x-5 \right )}{6}$$

Step 3: Solve the newly formed expression.

$$\frac{4x}{6}-\frac{9\left(x-5 \right )}{6}=\frac{4x-9x+45}{6}$$

$$\frac{4x}{6} -\frac{9\left(x-5 \right )}{6}=\frac{-5x+45}{6}$$

Step 4: Write the final answer.

$$\frac{4x}{6}-\frac{9\left(x-5 \right )}{6}=\frac{-5x+45}{6}$$

E2.3B: Factorise and simplify rational expressions

To simplify the rational expression, firstly, factorise both the numerator and the denominator individually. After factorisation, find the common factors between the numerator and the denominator and eliminate the like terms. Then, you have the simplified and factorised rational expression.

Worked examples

Example 1: Simplify $$\frac{x^{2}-9x-14}{x^{2}+2x-8}$$

Step 1: Write the given rational expression.

$$\frac{x^{2}-9x-14}{x^{2}+2x-8}$$

Step 2: Factorise the numerator of the given expression.

$$x^{2}-9x-14=x^{2}-2x-7x-14$$

$$x^{2}-9x-14=\left ( x-2 \right )\left ( x-7 \right )$$

Step 3: Factorise the denominator of the given expression.

$$x^{2}+2x-8=x^{2}-2x+4x-8$$

$$x^{2}+2x-8=x\left (x-2  \right )+4\left ( x-2 \right )$$

$$x^{2}+2x-8=\left ( x-2 \right )\left ( x+4 \right )$$

Step 4: Substitute the factorised values of numerator and denominator.

$$\frac{x^{2}-9x-14}{ x^{2}+2x-8}=\frac{\left ( x-2 \right )\left ( x-7 \right )}{\left ( x-2 \right )\left ( x+4 \right )}$$

Step 5: Simplify the above expression. 

$$\frac{\left ( x-2 \right )\left ( x-7 \right )}{\left ( x-2 \right )\left ( x-4 \right )}=\frac{\left ( x-7 \right )}{\left ( x-4 \right )}$$

Example 2: Simplify $$\frac{3x^{2}-9xy-12y^{2}}{6x^{3}-6xy^{2}}$$.

Step 1: Write the given rational expression.

$$\frac{3x^{2}-9xy-12y^{2}}{6x^{3}-6xy^{2}}$$

Step 2: Factorise the numerator of the given expression.

$$3x^{2}-9xy-12y^{2}=3x^{2}-12xy+3xy-12y^{2}$$

$$3x^{2}-9xy-12y^{2}=3x^{2}+3xy-12xy-12y^{2}$$

$$3x\left ( x+y \right )-12y\left ( x+y \right )$$

$$\left ( 3x-12y \right )\left ( x+y \right )$$

$$3\left ( x-4y \right )\left ( x+y \right )$$

Step 3: Factorise the denominator of the given expression.

$$6x^{3}-6xy^{2}=6x\left ( x^{2}-y^{2} \right )$$

$$6x^{3}-6xy^{2}=6x\left ( x+y \right )\left ( x-y \right )$$

Step 4: Substitute the factorised values of numerator and denominator.

$$\frac{3x^{2}-9xy-12y^{2}}{6x^{3}-6xy^{2}}=\frac{3\left ( x-4y \right )\left ( x+y \right )}{6x\left ( x+y \right )\left ( x-y \right )}$$

Step 5: Simplify the above expression.

$$\frac{3\left ( x-4y \right )\left ( x+y \right )} {6x\left ( x+y \right )\left ( x-y \right )}= \frac{\left ( x-4y \right )}{2x\left ( x-y \right )}$$