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ALGEBRAIC FRACTIONS

📌 Simplifying Algebraic Fractions

Algebraic fractions are simplified in the same way as numerical fractions are simplified:
  • Factorise the numerator and denominator into products of factors.

    • \[ \frac{15}{20} = \frac{3 \times 5}{4 \times 5} \]

  • Cancel common factors which appear in both the numerator and the denominator.

    • \[ \frac{3 \times \cancel{5}}{4 \times \cancel{5}} = \frac{3}{4} \]

Example 1: Simplify\( \frac{x^{2} - 4}{x^{2} - 5x + 6}\)

Step 1: Factorise \( \frac{ x^{2} - 4}{x^{2} - 5x + 6} = \frac{(x-2)(x+2)}{(x-2)(x-3)} \)

Step 2: Cancel common factors\( \frac{ \cancel{(x-2)}(x+2)}{\cancel{(x-2)}(x-3)} = \frac{ (x+2)}{ (x-3)}\)

Factorise then cancel
If you can't factorise, you can't cancel!
\( \frac{x^{2} - 4}{x^{2} - 5x +6} \neq \frac{\cancel{x^{2}} - 4}{\cancel{x^{2}} - 5x +6} \)

Practice

📌 Adding (or Subtracting) Algebraic Fractions

Algebraic fractions are added (or subtracted) in the same way as numerical fractions are added (or subtracted):
  • Factorise denominators.

    • \[ \frac{1}{12} + \frac{1}{15} = \frac{1}{3 \times 4} + \frac{1}{3 \times 5}\ \]

  • Convert fractions to equivalent fractions with the same denominators.

    • \[ \frac{1}{3 \times 4 } + \frac{1}{3 \times 5 } = \frac{1 \times 5 }{3 \times 4 \times 5} + \frac{1 \times 4}{3 \times 5 \times 4}\ = \frac{5}{60} + \frac{4}{60}\]

  • Add (or subtract) the numerators.

    • \[ \frac{5}{60} + \frac{4}{60} = \frac{9}{60} \]

  • Simplify the result if necessary.

    • \[ \frac{9}{60} = \frac{ \cancel{3} \times 3}{ \cancel{3} \times 20} = \frac{3}{20} \]

Example 2: Simplify: \( \frac{1}{x^{2} - 4} + \frac{1}{x^{2} - 5x + 6}\)

Step 1: Factorise the denominators: \( \frac{1}{x^{2} - 4} + \frac{1}{x^{2} - 5x + 6} = \frac{1}{(x-2)(x+2)} + \frac{1}{(x-2)(x-3)} \)

Step 2: Convert to equivalent fractions: \( \frac{1}{(x-2)(x+2)} + \frac{1}{(x-2)(x-3)} = \frac{1 \times(x-3) }{(x-2)(x+2) \times(x-3) } + \frac{1\times(x+2)}{(x-2)(x-3)\times(x+2) } \)

Step 3: Add (or subtract) numertors: \( \frac{(x-3) }{(x-2)(x+2)(x-3) } + \frac{(x+2)}{(x-2)(x-3)(x+2)} = \frac{(2x-1)}{(x-2)(x-3)(x+2)} \)

Practice

📌 Multipying Algebraic Fractions

Algebraic fractions are multiplied in the same way as numerical fractions are multiplied:
  • Factorise fractions.

    • \[ \frac{8}{15} \times \frac{21}{36} = \frac{2 \times 4}{3 \times 5} \times \frac{3 \times 7}{4 \times 9} \]

  • Cancel common factors between numerators and deonominators.

    • \[ \frac{2 \times \cancel{4}}{ \cancel{3} \times 5} \times \frac{ \cancel{3} \times 7}{ \cancel{4} \times 9} = \frac{2}{5} \times \frac{7}{9} \]

  • Multiply numerators and multiply denominators.

    • \[ \frac{ 2 \times 7}{5 \times 9} = \frac{14}{45} \]

Example 3: Simplify: \( \frac{x+3}{x^{2} - 4} \times \frac{3x - 6}{x^{2} + 7x + 12}\)

Step 1: Factorise the fractions: \( \frac{x+3}{x^{2} - 4} \times \frac{3x - 6}{x^{2} + 7x + 12} = \frac{(x + 3)}{(x-2)(x+2)} \times \frac{3(x-2)}{(x + 3)(x + 4)} \)

Step 2: Cancel common factors: \( = \frac{ \cancel{(x + 3)}}{\cancel{(x-2)}(x+2)} \times \frac{3\cancel{(x-2)}}{\cancel{(x + 3)}(x + 4)} = \frac{1}{(x+2)} \times \frac{3}{(x + 4)} \)

Step 3: Multiply numerators & multiply denominators: \( \frac{1 \times 3}{(x+2) \times (x+4)} = \frac{3}{(x+2)(x + 4)} \)

Practice

📌 Dividing Algebraic Fractions

Algebraic fractions are divided in the same way as numerical fractions are divided.

To divide by a fraction, convert the question into a multiplication question using:
  • Leave It, Change It, Flip It: Leave the first fraction, Change to multiplication, Flip the second fraction.
  • KFC: Keep, Flip, Change
  • Use Leave It - Change It - Flip It (or KFC)

    • \[ \frac{8}{15} \div \frac{36}{21} = \frac{8}{15} \times \frac{21}{36} \]

  • Factorise fractions.

    • \[ \frac{8}{15} \times \frac{21}{36} = \frac{2 \times 4}{3 \times 5} \times \frac{3 \times 7}{4 \times 9} \]

  • Cancel common factors between numerators and deonominators.

    • \[ \frac{2 \times \cancel{4}}{ \cancel{3} \times 5} \times \frac{ \cancel{3} \times 7}{ \cancel{4} \times 9} = \frac{2}{5} \times \frac{7}{9} \]

  • Multiply numerators and multiply denominators.

    • \[ \frac{ 2 \times 7}{5 \times 9} = \frac{14}{45} \]

Example 4: Simplify: \( \frac{x+3}{x^{2} - 4} \div \frac{x^{2} + 7x + 12}{3x - 6} \)

Step 1: Convert to multiplication & factorise: \( \frac{x+3}{x^{2} - 4} \times \frac{3x - 6}{x^{2} + 7x + 12} = \frac{(x + 3)}{(x-2)(x+2)} \times \frac{3(x-2)}{(x + 3)(x + 4)} \)

Step 2: Cancel common factors: \( = \frac{ \cancel{(x + 3)}}{\cancel{(x-2)}(x+2)} \times \frac{3\cancel{(x-2)}}{\cancel{(x + 3)}(x + 4)} = \frac{1}{(x+2)} \times \frac{3}{(x + 4)} \)

Step 3: Multiply numerators & multiply denominators: \( \frac{1 \times 3}{(x+2) \times (x+4)} = \frac{3}{(x+2)(x + 4)} \)

"The only way to learn mathematics is to do mathematics" - Paul R Halmos