Simplification of Fractions

An algebraic fraction is a fraction whose numerator or denominator or both numerator and denominator are algebraic expressions.

For example:

\( \frac{1}{5x}, \: \frac{2x}{5x + 6},\: \frac{x^2 \: + \: 5x}{(2x + 1)(3x + 5)}, \: \frac{a^2 \: + \: 2ab \: + \: b^2}{a^2\:-\:b^2}\scriptsize, \: etc \)

Example 10.1.1:

Simplify the following algebraic fractions:

a. \( \frac{x^5}{x^2} \)

b. \( \frac{60x^3y^7}{48x^5y^4}\)

c. \( \frac{2x^2 \: – \: 5x \: -\: 12}{4x^2 \: -\: 9}\)

d. \( \frac{4x^2 \: – \: 9}{2x^2 \: -\:5x \: + \: 3} \)

e. \( \frac{p^2 \: -\:pr \: -\: pq \: + \: qr}{p^2 \: + \: pr \: – \: pq \: – \: qr} \)

f. \( \frac{6 \: + \: x \: – \: x^2}{2x^2 \: + \: x \: – \: 6}\)

g. \( \frac{x^2 \: -\:16}{x^2 \: – \: 9x \: + \: 20} \)

Solution:

a. \( \frac{x^5}{x^2}\)

\( = \frac{x \: \times \: x \: \times \: x\: \times \: x \: \times \: x \: \times \: x}{x\: \times \:x} \\ = \frac{\not{x} \: \times \: \not{x} \: \times \: x\: \times \: x \: \times \: x \: \times \: x}{ \not{x}\: \times \: \not{x}} \\ = \scriptsize x \: \times \: x \: \times \: x \\ = x^3 \)

b. \( \frac{60x^3y^7}{48x^5y^4}\)

= \( \frac{60\: \times \: x \: \times \: x\: \times \: x \: \times \: y \: \times \: \\ y\: \times \: y\: \times \: y\: \times \: y\: \times \: y\: \times \: y}{48 \: \times \: x \: \times \: x \: \times \: x\: \times \: x\: \times \: x \\ \: \times \: y\: \times \: y\: \times \: y\: \times \: y} \)

= \( \frac{60\: \times \: \not{x} \: \times \: \not{x}\: \times \: \not{x} \: \times \: y \: \times \: \\ y\: \times \: y\: \times \: \not{y}\: \times \: \not{y}\: \times \: \not{y}\: \times \: \not{y}}{48 \: \times \: x \: \times \: x \: \times \: \not{x}\: \times \: \not{x}\: \times \: \not{x} \\ \: \times \: \not{y}\: \times \: \not{y}\: \times \: \not{y}\: \times \: \not{y}} \)

= \( \frac{60\: \times \:y^3}{48\: \times \: x^2} \)

Divide by 12

= \( \frac{5y^3}{4x^2} \)

or we can simply divide the numerator and denominator by their common factor 12x³y⁴

\( \frac{60x^3y^7}{48x^5y^4} \\ = \frac{60x^3y^7}{12x^3y^4} \: \div \: \frac{48x^5y^4}{12x^3y^4} \\ = \scriptsize 5y^3 \: \div \: 4x^2 \\ = \frac{5y^3}{4x^2} \)

c. \( \frac{2x^2 \: – \: 5x \: -\: 12}{4x^2 \: -\: 9}\)

factorise

= \( \frac{2x^2 \: – \: 8x\: + \: 3x \: -\: 12}{(2x)^2 \: -\: 3^2} \)

express (2x)² – 3² as difference of two squares