Multiplication and Division of Algebraic Fractions

Multiplication and Division of Algebraic Fractions#

Introduction#

Theory#

Multiplying and dividing algebraic fractions is actually simpler than addition and subtraction—you don’t need common denominators! These operations follow straightforward rules that mirror those for numerical fractions, with the added benefit that you can often simplify before multiplying.

Multiplication of Algebraic Fractions#

Theory#

The Basic Rule#

To multiply fractions, multiply the numerators together and multiply the denominators together:

\[\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} = \frac{ac}{bd}\]

The Smart Approach: Cancel First, Multiply Later#

The key to efficient multiplication is to cancel common factors before multiplying. This keeps numbers smaller and reduces the need for simplification later.

Tip

Always look for opportunities to cancel before multiplying. This is not just a time-saver—it also reduces the chance of arithmetic errors with large expressions.

Application#

Example 1: Basic Multiplication#

Multiply: \(\frac{x}{3} \times \frac{6}{x^2}\)

Method 1 (multiply first, then simplify):

\[\frac{x}{3} \times \frac{6}{x^2} = \frac{6x}{3x^2} = \frac{2}{x}\]

Method 2 (cancel first—more efficient):

\[\frac{x}{3} \times \frac{6}{x^2} = \frac{\cancel{x}}{3} \times \frac{6}{x^{\cancel{2}}} = \frac{1}{3} \times \frac{6}{x} = \frac{6}{3x} = \frac{2}{x}\]

Both methods work, but canceling first is cleaner!

Example 2: Factoring Before Multiplying#

Multiply: \(\frac{x^2-4}{x+3} \times \frac{x+3}{x-2}\)

Step 1: Factor where possible:

\[\frac{(x+2)(x-2)}{x+3} \times \frac{x+3}{x-2}\]

Step 2: Cancel common factors:

  • \((x+3)\) appears in numerator and denominator

  • \((x-2)\) appears in numerator and denominator

\[\frac{(x+2)\cancel{(x-2)}}{\cancel{x+3}} \times \frac{\cancel{x+3}}{\cancel{x-2}} = x+2\]

Domain: \(x \neq -3, x \neq 2\) (from the original fractions)

Note

Even though the final answer is simply \(x+2\), the domain restrictions from the original fractions still apply. This is crucial for exam success!

Example 3: Multiple Fractions#

Multiply: \(\frac{x^2-1}{x} \times \frac{2x}{x+1} \times \frac{3}{x-1}\)

Step 1: Factor all expressions:

\[\frac{(x+1)(x-1)}{x} \times \frac{2x}{x+1} \times \frac{3}{x-1}\]

Step 2: Write as one fraction:

\[\frac{(x+1)(x-1) \times 2x \times 3}{x \times (x+1) \times (x-1)}\]

Step 3: Cancel common factors:

\[\frac{\cancel{(x+1)}\cancel{(x-1)} \times 2\cancel{x} \times 3}{\cancel{x} \times \cancel{(x+1)} \times \cancel{(x-1)}} = 2 \times 3 = 6\]

Division of Algebraic Fractions#

Theory#

The Basic Rule#

To divide by a fraction, multiply by its reciprocal:

\[\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}\]

Remember: “Dividing by a fraction = multiplying by its flip”

Warning

A common error is to flip the wrong fraction. Always flip the fraction you’re dividing BY (the second one), not the fraction you’re dividing.

Application#

Example 4: Simple Division#

Divide: \(\frac{x^2}{4} \div \frac{x}{2}\)

Step 1: Rewrite as multiplication by the reciprocal:

\[\frac{x^2}{4} \div \frac{x}{2} = \frac{x^2}{4} \times \frac{2}{x}\]

Step 2: Cancel and multiply:

\[= \frac{x^{\cancel{2}}}{4} \times \frac{2}{\cancel{x}} = \frac{x}{4} \times \frac{2}{1} = \frac{2x}{4} = \frac{x}{2}\]

Example 5: Division with Factoring#

Divide: \(\frac{x^2-9}{x^2+x} \div \frac{x-3}{x}\)

Step 1: Factor and flip:

\[\frac{(x+3)(x-3)}{x(x+1)} \div \frac{x-3}{x} = \frac{(x+3)(x-3)}{x(x+1)} \times \frac{x}{x-3}\]

Step 2: Cancel common factors:

\[= \frac{(x+3)\cancel{(x-3)}}{\cancel{x}(x+1)} \times \frac{\cancel{x}}{\cancel{x-3}} = \frac{x+3}{x+1}\]

Domain: \(x \neq 0, x \neq -1, x \neq 3\)

Example 6: Complex Division#

Simplify: \(\frac{\frac{x^2-4}{x}}{\frac{x+2}{x^2}}\)

This is a division of two fractions. We can rewrite it as:

\[\frac{x^2-4}{x} \div \frac{x+2}{x^2}\]

Step 1: Factor and flip:

\[\frac{(x+2)(x-2)}{x} \times \frac{x^2}{x+2}\]

Step 2: Cancel:

\[= \frac{\cancel{(x+2)}(x-2)}{\cancel{x}} \times \frac{x^{\cancel{2}}}{\cancel{x+2}} = (x-2) \times x = x(x-2) = x^2-2x\]

Mixed Operations#

Theory#

When combining multiplication and division, work from left to right, converting each division to multiplication by the reciprocal.

Application#

Example 7: Mixed Operations#

Simplify: \(\frac{x}{x-1} \times \frac{x^2-1}{x^2} \div \frac{x+1}{x}\)

Step 1: Convert division to multiplication:

\[\frac{x}{x-1} \times \frac{x^2-1}{x^2} \times \frac{x}{x+1}\]

Step 2: Factor:

\[\frac{x}{x-1} \times \frac{(x+1)(x-1)}{x^2} \times \frac{x}{x+1}\]

Step 3: Combine into one fraction:

\[\frac{x \times (x+1)(x-1) \times x}{(x-1) \times x^2 \times (x+1)}\]

Step 4: Cancel:

\[\frac{\cancel{x} \times \cancel{(x+1)}\cancel{(x-1)} \times x}{\cancel{(x-1)} \times x^{\cancel{2}} \times \cancel{(x+1)}} = \frac{x}{x} = 1\]

Domain: \(x \neq 0, x \neq 1, x \neq -1\)

Special Techniques and Patterns#

Technique 1: Recognizing Conjugates#

When you see expressions like \((a+b)\) and \((a-b)\), their product is \(a^2-b^2\).

Example: \(\frac{x+\sqrt{3}}{x-\sqrt{3}} \times \frac{x-\sqrt{3}}{x+\sqrt{3}} = \frac{(x+\sqrt{3})(x-\sqrt{3})}{(x-\sqrt{3})(x+\sqrt{3})} = 1\)

Technique 2: Powers of Fractions#

Remember that \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

Example: \(\left(\frac{x-1}{x+1}\right)^2 = \frac{(x-1)^2}{(x+1)^2}\)

Technique 3: Negative Exponents#

Recall that \(x^{-n} = \frac{1}{x^n}\), so:

\[\frac{x^{-2}}{y^{-3}} = \frac{1/x^2}{1/y^3} = \frac{1}{x^2} \times \frac{y^3}{1} = \frac{y^3}{x^2}\]

Common Mistakes to Avoid#

Mistake 1: Canceling Addition/Subtraction#

❌ Wrong: \(\frac{x+2}{x} = 2\)

✓ Correct: \(\frac{x+2}{x} = \frac{x}{x} + \frac{2}{x} = 1 + \frac{2}{x}\)

Only factors can be canceled, not terms!

Mistake 2: Forgetting to Flip When Dividing#

❌ Wrong: \(\frac{a}{b} \div \frac{c}{d} = \frac{ac}{bd}\)

✓ Correct: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}\)

Mistake 3: Domain Restrictions#

Even if factors cancel, the original restrictions remain.

If you simplify \(\frac{x^2-1}{x-1}\) to \(x+1\), you must still note that \(x \neq 1\).

Problem-Solving Strategy#

  1. Factor everything first - This reveals cancelation opportunities

  2. Convert division to multiplication - Flip the divisor

  3. Cancel before multiplying - This keeps expressions manageable

  4. State domain restrictions - Include all values that make any denominator zero

  5. Check your answer - Substitute a simple value to verify

Tip

When checking your work, choose a simple test value (like \(x = 2\) or \(x = 0\)) that doesn’t violate any domain restrictions. Evaluate both the original expression and your answer—they should give the same result.

Applications#

Multiplication and division of algebraic fractions appear in:

  • Rate problems: Distance/time, work rates, flow rates

  • Physics formulas: Resistance in parallel circuits, lens equations

  • Economics: Compound interest, depreciation formulas

  • Calculus: Derivative and integration techniques

Summary#

For multiplication:

  • Multiply numerators and denominators

  • Factor and cancel first for efficiency

  • Watch for domain restrictions

For division:

  • Flip the second fraction and multiply

  • Remember: “keep, change, flip”

  • Apply the same factoring and canceling techniques

These operations are generally easier than addition/subtraction because you don’t need common denominators—just factor, cancel, and multiply!

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