Collecting, expanding and factorising expressions · Higher
| English | 中文 | Pinyin |
|---|---|---|
| factorise/ˈfæktəraɪz/ | 因式分解 | yīn shì fēn jiě |
Can two area descriptions agree?
- An L-shaped area has a symbolic width. Can two different expressions describe the same total area?
- This lesson studies factorise 因式分解: Rewrite an expression as a product of factors.
Choose the mathematical structure
- Collect only like terms. Distribute multiplication over every term in a bracket, including signs. Factorising reverses expansion. Use common factors first; then pairs for a quadratic. Algebraic fractions may cancel common factors, with forbidden denominator values stated.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines factorise?
Rewrite an expression as a product of factors.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
3x+4x=7x, but 3x+4y cannot be combined. Expand 2(x+3)=2x+6 and (x+2)(x+3)=x²+5x+6. Reverse the last result to factorise x²+5x+6. Difference of squares gives x²-9=(x-3)(x+3). For Higher, 2x²+5x+2=(2x+1)(x+2), and (x²-9)/(x-3)=x+3 only when x≠3. Also (x+1)(x+2)(x+3)=x³+6x²+11x+6.
Collecting, expanding and factorising expressions
Collect only like terms
Compare the model with the worked case and explain one change.
Find the coefficient of x in 3x+4x.
Add coefficients 3+4.
Test a tempting shortcut
- x² and x are unlike terms. Cancelling across a sum is invalid; factor the whole expression first. A simplified fraction must retain values excluded by its original denominator.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A term can be cancelled from a numerator and denominator even if it is not a factor of the whole expression. This claim is false. Explain which definition or assumption it violates.
Find the constant term after expanding 2(x+3).
Distribute 2 to both terms; 2×3=6.
A term can be cancelled from a numerator and denominator even if it is not a factor of the whole expression.
x² and x are unlike terms. Cancelling across a sum is invalid; factor the whole expression first. A simplified fraction must retain values excluded by its original denominator.
Interpret a new situation
- AQA A4 Foundation includes collecting, single brackets, common factors, two-binomial expansion and monic quadratic factorisation. Higher adds non-monic factorisation, more binomials, surds and algebraic fractions.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the coefficient of x in (x+2)(x+3).
The cross terms are 3x+2x=5x.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.2. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
Rewrite an expression as a product of factors. Choose the relationship, show the method, check its assumptions and interpret the result.