Collecting, expanding and factorising expressions · Foundation
| 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 like terms, distribute over brackets and reverse expansion by factorising. Take out a common factor first. Expand products of two binomials and factorise simple monic quadratics and differences of squares.
- 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; 3x+4y cannot be combined. Expand 2(x+3)=2x+6 and (x+2)(x+3)=x²+5x+6. Reverse the last identity to factorise x²+5x+6. Difference of squares gives x²-9=(x-3)(x+3). The same expansion rule works with given roots: (√2+1)(√2-1)=2-1=1.
Collecting, expanding and factorising expressions
Collect like terms, distribute over brackets and reverse expansion by factorising
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. A factor multiplies a whole expression; it is not a separate added term. Check a proposed factorisation by expanding it.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The terms 3x and 4y can be collected to give 7xy. 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.
The terms 3x and 4y can be collected to give 7xy.
x² and x are unlike terms. A factor multiplies a whole expression; it is not a separate added term. Check a proposed factorisation by expanding it.
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 · Foundation · 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.