Algebraic ManipulationEdexcel GCSE Maths: Revision notes
Section 1
How do you simplify algebraic expressions by collecting like terms?
Like terms are terms that contain the same variable raised to the same power. To simplify, identify these terms and combine them using addition or subtraction.
Process:
- Identify all like terms (same letter, same power)
- Group them together
- Add or subtract the coefficients
- Write the simplified expression
Examples of like and unlike terms:
| Expression | Like Terms | Unlike Terms |
|---|---|---|
| 3x + 5x + 2y | 3x and 5x (= 8x) | 3x and 2y |
| 4a² + 2a + 3a² | 4a² and 3a² (= 7a²) | a² and a |
| 6xy + 2y + xy | 6xy and xy (= 7xy) | xy and y |
Worked example: Simplify: 4x + 3y - 2x + 5y + 1
- Group like terms: (4x - 2x) + (3y + 5y) + 1
- Combine: 2x + 8y + 1
Examiners expect you to show all grouping. Write out the regrouped terms explicitly before combining—this demonstrates clear algebraic thinking.
Students often add coefficients without checking powers match. Remember: 2x and 2x² are NOT like terms because the powers differ.
Section 2
How do you expand single, double and triple brackets?
Single brackets: Multiply each term inside by the term outside.
Worked example (single): 3(2x + 5) = 3 × 2x + 3 × 5 = 6x + 15
Double brackets: Use FOIL (First, Outer, Inner, Last) or the grid method. Multiply each term in the first bracket by each term in the second.
Worked example (double): (x + 2)(x + 3)
- First: x × x = x²
- Outer: x × 3 = 3x
- Inner: 2 × x = 2x
- Last: 2 × 3 = 6
- Combine: x² + 5x + 6
Triple brackets: Expand two brackets first, then expand the result with the third bracket.
Worked example (triple): (x + 1)(x + 2)(x + 3)
- Expand (x + 1)(x + 2) = x² + 3x + 2
- Expand (x² + 3x + 2)(x + 3) = x³ + 3x² + 3x² + 9x + 2x + 6 = x³ + 6x² + 11x + 6
For double brackets, use a grid method if FOIL confuses you—it makes tracking each multiplication clear and reduces careless errors.
Forgetting the negative sign when expanding. For (x - 3)(x + 2), the inner term is -3 × x = -3x, not +3x.
Expanding brackets is like distributing parcels to multiple houses: each item inside must go to each house outside.
Section 3
How do you factorise expressions by taking out a common factor?
Factorisation is the reverse of expansion. Find the greatest common factor (GCF) of all terms, then write it outside brackets.
Process:
- Identify the GCF of all terms (including numbers and variables)
- Divide each term by the GCF
- Write the GCF outside brackets with the divided terms inside
Worked example: Factorise: 6x² + 9x
- GCF of 6x² and 9x is 3x
- 6x² ÷ 3x = 2x; 9x ÷ 3x = 3
- Answer: 3x(2x + 3)
- Check: 3x × 2x + 3x × 3 = 6x² + 9x ✓
Another example: Factorise: 12a³b + 8ab²
- GCF is 4ab
- 12a³b ÷ 4ab = 3a²; 8ab² ÷ 4ab = 2b
- Answer: 4ab(3a² + 2b)
Always check your factorisation by expanding the brackets back. This catches mistakes and shows examiners your work is correct.
Taking out only partial common factors. If all terms contain x, make sure you include x in the GCF, not just numbers.
Section 4
How do you factorise quadratic expressions?
Quadratic expressions are of the form ax² + bx + c. Factorisation methods depend on the values of a, b, and c.
Difference of two squares: x² - a² = (x + a)(x - a)
Worked example:
- x² - 25 = (x + 5)(x - 5)
- 4x² - 9 = (2x + 3)(2x - 3)
Simple quadratics (a = 1): x² + bx + c = (x + p)(x + q) where p + q = b and pq = c
Worked example: x² + 7x + 12
- Find two numbers that add to 7 and multiply to 12: 3 and 4
- Answer: (x + 3)(x + 4)
- Check: x² + 4x + 3x + 12 = x² + 7x + 12 ✓
General quadratics (a ≠ 1, Higher Tier): ax² + bx + c
Method:
- Find two numbers that multiply to ac and add to b
- Split the bx term using these numbers
- Factorise by grouping
Worked example: 2x² + 7x + 3
- ac = 2 × 3 = 6; b = 7
- Two numbers: 6 and 1 (6 × 1 = 6, 6 + 1 = 7)
- 2x² + 6x + 1x + 3
- Group: (2x² + 6x) + (1x + 3)
- Factorise: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)
For difference of two squares, check both terms are perfect squares. x² - 25 works because 25 = 5², but x² + 25 cannot be factorised this way.
With ax² + bx + c, students forget to include a in the ac product. For 2x² + 5x + 2, ac = 4, not just c.
Section 5
How do you simplify and perform operations with algebraic fractions? (Higher Tier)
Algebraic fractions are fractions containing variables. Simplify by factorising the numerator and denominator, then cancelling common factors.
Simplifying:
Worked example: Simplify: (3x + 6)/(x + 2)
- Factorise numerator: 3(x + 2)/(x + 2)
- Cancel common factor: 3
Simplify: (x² - 4)/(x + 2)
- Factorise numerator: (x + 2)(x - 2)/(x + 2)
- Cancel: x - 2
Adding and subtracting fractions: Find a common denominator, combine numerators, simplify.
Worked example: 3/x + 2/(x + 1)
- Common denominator: x(x + 1)
- 3(x + 1)/[x(x + 1)] + 2x/[x(x + 1)] = [3x + 3 + 2x]/[x(x + 1)] = (5x + 3)/[x(x + 1)]
Multiplying fractions: Multiply numerators and denominators; factorise and cancel before multiplying if possible.
Dividing fractions: Flip the second fraction and multiply.
Worked example (multiply): (x + 2)/(x - 1) × (x - 1)/(x + 3)
- Cancel (x - 1): (x + 2)/(x + 3)
Worked example (divide): (2x)/(x + 3) ÷ (4x)/(x + 1) = (2x)/(x + 3) × (x + 1)/(4x)
- Cancel 2x: (x + 1)/[2(x + 3)]
Always factorise before cancelling. Never cancel terms—only factors. For example, in (x + 2)/(x + 3), you cannot cancel the x's because they are not factors.
Cancelling incorrectly. (2x + 4)/(2) = (x + 2), not x + 4. You must factorise the numerator first: 2(x + 2)/2 = x + 2.
Must Know
- Like terms have identical variable parts (same letters to same powers); combine their coefficients to simplify expressions.
- Expand brackets by multiplying every term inside by every term outside; use FOIL or grid method for double brackets.
- Factorise by finding the greatest common factor (GCF) of all terms, dividing each term by it, and writing the GCF outside the brackets; always verify by expanding.
- Difference of two squares x² - a² = (x + a)(x - a); recognise perfect square terms to use this method.
- Quadratic factorisation (ax² + bx + c) requires finding two numbers that multiply to ac and add to b; use these to split the bx term, then factorise by grouping.
- Algebraic fractions are simplified by factorising numerator and denominator, then cancelling common factors; for operations, find common denominators (add/subtract) or flip and multiply (divide); always verify answers cannot be simplified further.
That's the notes covered.
Carry on to the next subtopic.