Playbook
Algebraic Identities
98 q · 39% HARD · the hardest chapter in the bank, about 4.7 a paper. It cherry-picks: conditional identities (69% HARD) and symmetric expressions (6 of 6 HARD) are the pool; three-variable sums and the cube identity are cheap.
- Questions in the bank
- 98
- q/paper (21 papers)
- 4.67
- Tagged HARD
- 39%
- Subtopics
- 8
Strand: Selective
When you’ll see it
An expression to simplify, or a value of one symmetric expression given and another asked.
How this chapter is tested
98 questions, about 4.7 a paper, and the hardest chapter in the bank at 39% HARD. It cherry-picks: conditional identities (16 questions, 69% HARD) and symmetric and cyclic expressions (6 of 6 HARD) are the pool.
The cheap pages are the standard expansions and the cube identity: if a + b + c = 0 then a³ + b³ + c³ = 3abc. When a question gives values, substituting simple numbers that satisfy the condition is often the fastest check.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Squares and cubes of a binomial
(a ± b)² and (a ± b)³, and working backwards from a sum and a product.
Reciprocal sums
From x + 1/x = k: x² + 1/x² = k² − 2 and x³ + 1/x³ = k³ − 3k.
Three variables
(a + b + c)² = Σa² + 2Σab; the cube identity and a + b + c = 0.
Least values
A sum of squares is least when each square is zero.
Conditional identities
Use the given condition to replace one variable, or test with numbers that satisfy it.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.
x² + 1/x² from x + 1/x
It is k² − 2, not k².
a + b + c = 0 not given
Without it, a³ + b³ + c³ − 3abc is (a + b + c)(Σa² − Σab), not zero.
Learn it before you drill it
This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each subtopic. Read the notes once, then drill subtopic by subtopic below.
Algebraic Identities notesDrill every Algebraic Identities question
98 questions from the bank, across 8 subtopics.
Drill one subtopic at a time
The 8 subtopics, in teaching order.
- Sums and Products of Three VariablesDrill Sums and Products of Three Variables
- The Cube Identity and a + b + c = 0Drill The Cube Identity and a + b + c = 0
- Squares and Cubes of a BinomialDrill Squares and Cubes of a Binomial
- Reciprocal Sums x ± 1/xDrill Reciprocal Sums x ± 1/x
- Rational Algebraic ExpressionsDrill Rational Algebraic Expressions
- Sums of Squares and Least ValuesDrill Sums of Squares and Least Values
- Conditional IdentitiesDrill Conditional Identities
- Symmetric and Cyclic ExpressionsDrill Symmetric and Cyclic Expressions
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