Algebraic Fractions

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Notes

Simplifying Algebraic Fractions

**Factorise** numerator and denominator fully.
Cancel any **common factors** (single term or bracket).
Cannot cancel a factor unless it is common to **all terms** in numerator/denominator.
E.g. $\frac{6x}{x+1}$ cannot be simplified.
Always check if numerator factorises after simplification.

Adding & Subtracting Algebraic Fractions

Find the **lowest common denominator (LCD)**.
LCD may be product of denominators, e.g. $(x+2)(x+5)$.
If denominators share a factor, LCD is the **LCM**, e.g. $\frac{1}{x}$ and $\frac{1}{2x}$ have LCD $2x$.
Rewrite each fraction over the LCD, multiplying numerator accordingly.
Combine into a single fraction and **simplify numerator**.
Check if numerator factorises for further cancellation.

Multiplying Algebraic Fractions

**Factorise** all numerators and denominators first.
Cancel any common factors **between any numerator and any denominator**.
Multiply numerators together and denominators together.
Check for further simplification.

Dividing Algebraic Fractions

**Flip** the second fraction (find its reciprocal) and change $\div to \times$ .
Then follow the rules for multiplying algebraic fractions.
E.g. $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$.

Solving Equations with Algebraic Fractions

Method 1: Combine fractions into a single fraction, then **cross-multiply** and solve.
Method 2: Multiply **every term** by each denominator to clear fractions.
Use brackets when multiplying by algebraic expressions.
Solve the resulting polynomial equation (often quadratic).

Common Mistakes & Tips

Always factorise before cancelling or adding fractions.
When subtracting, be careful with **negative signs** across brackets.
Check for **difference of two squares** (e.g. $x²-25$).
Leave answer in factorised form to see if cancellation is possible.

Simplifying Algebraic Fractions

Adding Algebraic Fractions

Multiplying Algebraic Fractions

Solving Equations with Algebraic Fractions

Practice questions

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1.Simplify $\frac{5}{3x} \times \frac{9x}{20}$ .
Easy
- A $\frac{3}{4}$
- B $\frac{45x^2}{60x}$
- C $\frac{3x}{4}$
- D $\frac{45}{60}$
2.Simplify $\frac{x^2+5x}{x^2-25}$ .
Easy
- A $\frac{x}{x-5}$
- B $\frac{x+5}{x-5}$
- C $\frac{x}{x+5}$
- D $\frac{x+5}{x+5}$
3.Simplify $\frac{x^2-25}{x^2-2x-35}$ .
Easy
- A $\frac{x-5}{x-7}$
- B $\frac{x+5}{x+7}$
- C $\frac{x-5}{x+7}$
- D $\frac{x+5}{x-7}$
4.Simplify $\frac{5p^2-20}{p} \div \frac{2p^2-32}{p}$ .
Medium
- A $\frac{5}{2}$
- B $\frac{5(p^2-4)}{2(p^2-16)}$
- C $\frac{5(p-2)}{2(p-4)}$
- D $\frac{5(p+2)}{2(p+4)}$
5.Write $\frac{x-5}{3} + \frac{6}{x+2}$ as a single fraction in its simplest form.
Medium
- A $\frac{x^2-3x-10+18}{3(x+2)}$
- B $\frac{x^2-3x-10}{3(x+2)}$
- C $\frac{x^2-3x+8}{3(x+2)}$
- D $\frac{x^2-3x-10+6}{3(x+2)}$
6.Simplify $\frac{p}{2q} \times \frac{4pq}{t}$ .
Medium
- A $\frac{2p^2}{t}$
- B $\frac{4p^2q}{2qt}$
- C $\frac{2p^2q}{t}$
- D $\frac{4p^2}{t}$
7.Simplify $\frac{ab-b^2}{a^2-b^2}$ .
Medium
- A $\frac{b}{a+b}$
- B $\frac{b}{a-b}$
- C $\frac{a-b}{a+b}$
- D $\frac{a}{a+b}$
8.Simplify $\frac{x^2-5x}{2x^2-50}$ .
Hard
- A $\frac{x}{2(x+5)}$
- B $\frac{x}{2(x-5)}$
- C $\frac{x-5}{2(x+5)}$
- D $\frac{x}{2x+10}$

Summary

## Algebraic Fractions - Night Before Exam Summary ### Simplifying - **Factorise** numerator and denominator fully. - **Cancel** common factors (single term or bracket). - Example: $\frac{4x+6}{2x^2-7x-15} = \frac{2(2x+3)}{(2x+3)(x-5)} = \frac{2}{x-5}$ ### Adding & Subtracting 1. Find **lowest common denominator (LCD)**. 2. Write each fraction over LCD. 3. Combine numerators, simplify. 4. Check if final fraction can be simplified. - Example: $\frac{x}{x+4} - \frac{3}{x-1} = \frac{x(x-1)-3(x+4)}{(x+4)(x-1)} = \frac{x^2-4x-12}{(x+4)(x-1)} = \frac{(x+2)(x-6)}{(x+4)(x-1)}$ ### Multiplying 1. Factorise all numerators and denominators. 2. Cancel common factors across fractions. 3. Multiply tops together, bottoms together. - Example: $\frac{x}{3x+6} \times \frac{2x+4}{x+7} = \frac{x}{3(x+2)} \times \frac{2(x+2)}{x+7} = \frac{2x}{3(x+7)}$ ### Dividing - Flip the second fraction (reciprocal) and multiply. - Then follow multiplication steps. - Example: $\frac{x+3}{x-4} \div \frac{2x+6}{x^2-16} = \frac{x+3}{x-4} \times \frac{(x-4)(x+4)}{2(x+3)} = \frac{x+4}{2}$ ### Solving Equations **Method 1:** Combine fractions into a single fraction, then cross-multiply and solve. **Method 2:** Multiply every term by all denominators to clear fractions, then solve. - Example: $\frac{4}{x-3} + \frac{5}{x+1} = 5$ → multiply by $(x-3)(x+1)$ → $4(x+1)+5(x-3)=5(x-3)(x+1)$ → solve quadratic. ### Key Tips - Always factorise first. - Watch for difference of two squares: $a^2-b^2 = (a-b)(a+b)$. - Be careful with negative signs when subtracting. - Check if final answer can be simplified further.

Key terms

Algebraic fraction: A fraction with an algebraic expression in the numerator and/or denominator.
Lowest common denominator (LCD): The smallest expression that is a multiple of all denominators in a set of fractions.
Reciprocal: The fraction obtained by swapping the numerator and denominator; used for division.
Factorising: Writing an expression as a product of its factors (e.g., common factor, quadratic factorisation, difference of two squares).
Cross-multiplying: Multiplying both sides of an equation by the denominators to eliminate fractions.

Slides

Simplifying Algebraic Fractions

Factorise the numerator and denominator fully.
Cancel any common factors (single terms or brackets).
A common factor must be common to all terms in the numerator or denominator.

Simplify: $(x^{2} + 5x) / (x^{2} - 25)$

x/(x-5)
(x+5)/(x-5)
x/(x+5)
5/(x-5)

Adding & Subtracting Algebraic Fractions

Find the lowest common denominator (LCD).
Rewrite each fraction with the LCD.
Add or subtract the numerators, then simplify if possible.

The LCD of 1/(x+2) and 1/(x+5) $is ____$ .

Multiplying Algebraic Fractions

Factorise numerators and denominators first.
Cancel any common factors across the fractions.
Multiply the numerators together and the denominators together.

Multiply: $(x)/(3x+6) \times (2x+4)/(x+7)$

(2x)/(3(x+7))
(2x)/(3x+7)
(x)/(3(x+7))
(2x)/(x+7)

Dividing Algebraic Fractions

Flip the second fraction (find its reciprocal).
Change division to multiplication.
Then follow the rules for multiplying fractions.

To divide $(x+3)/(x-4) \div (2x+6)/(x^{2}-16)$ , first rewrite as $(x+3)/(x-4) \times ____$ .

Solving Equations with Algebraic Fractions

Method 1: Combine fractions into a single fraction, then cross-multiply.
Method 2: Multiply every term by each denominator to clear fractions.
Always check for extraneous solutions (denominators cannot be zero).

Solve: $4x/(5x-1) = \frac{8}{7}$

$x = 2$
$x = \frac{1}{2}$
$x = -2$
$x = 4$

Worked Example: Simplifying

Simplify $(4x+6)/(2x^{2}-7x-15)$ .
Factorise numerator: 2(2x+3).
Factorise denominator: (2x+3)(x-5).
Cancel (2x+3) to get 2/(x-5).

Simplify $(x^{2} - 25)/(x^{2} - 2x - 35)$ . The numerator factorises to (x-5)(x+5). The denominator factorises to (x-7)(x+5). After cancelling, the answer $is ____$ .

Worked Example: Adding

Express $x/(x+4) - 3/(x-1)$ as a single fraction.
LCD: (x+4)(x-1).
Rewrite: $[x(x-1) - 3(x+4)] / [(x+4)(x-1)]$ .
Simplify numerator: $x^{2} - 4x - 12 = (x+2)(x-6)$ .

What is the LCD of 1/(x+2) and 1/((x+2)(x-1))?

(x+2)(x-1)
$(x+2)^{2}(x-1)$
(x+2)
$(x+2)(x-1)^{2}$

Worked Example: Multiplying

Multiply $(2x^{2}-11x+12)/(x^{2}+3x-28) \times (x^{2}-5x-14)/(9-4x^{2})$ .
Factorise each quadratic and difference of squares.
Cancel common factors, including rewriting 2x-3 as -(3-2x).
Final answer: -(x+2)(x-7)/[(x+7)(3+2x)].

Which of these are true about simplifying algebraic fractions?

Factorise numerator and denominator first.
Cancel any term that appears in both numerator and denominator.
A common factor must be common to all terms in the numerator or denominator.
You can cancel x in (6x)/(x+1).

Worked Example: Solving

Solve $2/(p+3) - \frac{5}{p} = 6p$ .
Multiply by $(p+3): 2 - 5(p+3)/p = 6p(p+3)$ .
Multiply by $p: 2p - 5(p+3) = 6p^{2}(p+3)$ .
Expand and rearrange to $6p^{3} + 18p^{2} + 3p + 15 = 0$ .

Solve: $1/(x-2) - 2/(x+1) = 3$ . After combining fractions and cross-multiplying, which equation do you get?

$3x^{2} + 3x - 5 = 0$
$3x^{2} + 3x + 5 = 0$
$3x^{2} - 3x - 5 = 0$
$3x^{2} - 3x + 5 = 0$

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