Identities — proving algebraic equivalence
An identity is an equation that holds true for every value of the variable. To prove one, you transform one side until it matches the other — you never assume the result is true and "balance" both sides like an equation.
How identities differ from equations
- Equation:
2x + 4 = 10→ solve to find x = 3. - Identity:
2(x + 2) ≡ 2x + 4→ no solving; both sides describe the same expression.
The symbol ≡ ("identically equal to") signals an identity. In examination questions you might also see "Show that..." or "Prove that...".
The proof technique
Pick the more complicated side (usually the LHS) and transform it step by step into the other side. Show every step. Never start from "what we are trying to prove".
✦Worked example— Worked example 1 — straight expansion
Show that (x + 4)(x - 4) ≡ x² - 16.
LHS = (x + 4)(x - 4) = x² - 4x + 4x - 16 = x² - 16 = RHS. □
✦Worked example— Worked example 2 — perfect square
Show that (2n + 1)² ≡ 4n² + 4n + 1.
LHS = (2n + 1)² = 4n² + 2n + 2n + 1 = 4n² + 4n + 1 = RHS. □
✦Worked example— Worked example 3 — comparing coefficients
Given (x + a)(x + b) ≡ x² + 7x + 12 for all x, find a and b.
Expand: x² + (a + b)x + ab. Compare:
- Coefficient of x:
a + b = 7. - Constant:
ab = 12.
So a and b are 3 and 4 (either way round).
✦Worked example— Worked example 4 — proof with consecutive integers
Show that the sum of three consecutive integers is always a multiple of 3.
Let the integers be n - 1, n, n + 1. Sum = (n - 1) + n + (n + 1) = 3n, which is a multiple of 3. □
This consecutive-integer trick is examined every year. Pick a clever centre (n) so symmetry kills the constants.
✦Worked example— Worked example 5 — algebraic divisibility
Show that (2n + 1)² - (2n - 1)² is always a multiple of 8.
Use difference of two squares: (2n + 1 + 2n - 1)(2n + 1 - 2n + 1) = (4n)(2) = 8n. So the expression is exactly 8n, a multiple of 8. □
⚠Common mistakes— Common mistakes (examiner traps)
- Starting from the conclusion. Assuming what you need to prove and "balancing" both sides earns 0 marks. Always work from one side to the other.
- Sign errors during expansion. Especially with perfect squares:
(a - b)² = a² - 2ab + b², nota² - b². - Forgetting the symmetry trick for "any/all integers" proofs. Choose your variables wisely (e.g.
n - 1, n, n + 1for three consecutives). - Comparing constants but forgetting to compare x-coefficients when matching identities.
- Stopping before the answer matches the RHS. Show the final simplification step.
➜Try this— Quick check
Show that (n + 3)² - (n + 1)² ≡ 4n + 8.
LHS = (n² + 6n + 9) - (n² + 2n + 1) = 4n + 8. □
AI-generated · claude-opus-4-7 · v3-deep-algebra