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Year 7: Simplification and Expansion of Algebraic Expressions (Advanced)

Quick Reference

Like Terms: Exact same variables with exact same powers
Expand: Distribute across all terms, watch signs carefully
Factor: Reverse of expanding; find common factors

Advanced Simplification Techniques

In advanced expressions, be systematic about grouping like terms.

Example with mixed terms:

$5 x^{2} - 3 x + 2 - 2 x^{2} + 4 x - 7$

Group by power: $(5 x^{2} - 2 x^{2}) + (- 3 x + 4 x) + (2 - 7)$
Combine: $3 x^{2} + x - 5$

Expanding Complex Expressions

With multiple variables and powers, expand methodically:

$2 x (3 x + 4 y - 5) = 6 x^{2} + 8 x y - 10 x$ (multiply 2x by each term)
$- 3 (2 a^{2} - a b + 1) = - 6 a^{2} + 3 a b - 3$ (distribute negative)
$a (a + b + c) = a^{2} + a b + a c$ (variable multiplication)

Combining Expansion and Simplification

Multi-step problems often require both skills:

Example: $2 (3 x + 2) + 4 (x - 1)$

Expand first: $6 x + 4 + 4 x - 4$
Simplify: 10x

Introduction to Factoring

Factoring is the reverse of expanding. Find the common factor:

$2 x + 4 = 2 (x + 2)$ (common factor is 2)
$3 a b + 6 a = 3 a (b + 2)$ (common factor is 3a)
$x^{2} + 3 x = x (x + 3)$ (common factor is x)

Key Insight: When you multiply out factored forms, you should get back the original expression. Use this to check your work!

Common Mistakes:

Forgetting to change signs when distributing negatives
Not identifying the greatest common factor when factoring
Leaving out terms or changing their signs in multi-bracket expansions