Exploring Algebraic Identities

What is an identity?

An equation like 2x + 1 = 7 is true only for x = 3. An identity is true for every value. Example: (x + 1)² = x² + 2x + 1. Try x = 5: 36 = 25 + 10 + 1. ✓

To test an identity, put any number in both sides. To prove it, expand or draw it.

Seeing identities as areas

Area of a rectangle = length × breadth. So a product of two brackets is an area.

• (a + b)² = a² + 2ab + b²: a square of side a + b = one a², two ab, one b².
• (a − b)² = a² − 2ab + b²: from a², remove two a×b strips. The b² corner was removed twice, so add it back once.
• a² − b² = (a + b)(a − b): cut b² from a corner of a². Move one piece of the L-shape to make a rectangle (a + b) by (a − b).
• (x + a)(x + b) = x² + (a + b)x + ab: rectangle with sides x + a and x + b.

Factorisation using identities and algebra tiles

Factorising means writing an expression as a product. Read the identity from right to left.

• x² + 10x + 25 = (x + 5)² (first and last are squares, middle is 2 × x × 5).
• 49y² − 16 = (7y)² − 4² = (7y + 4)(7y − 4).
• x² + 7x + 12: find two numbers with sum 7 and product 12 → 3 and 4. So (x + 3)(x + 4).

Algebra tiles: take one big square for x², a strip for each x, and a small square for each 1. Put the x² in a corner, the strips along two sides and fill the corner with units. If they make a perfect rectangle, its sides are the factors.

Finding new identities

We can build new identities from old ones or from shapes.

• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca: a square of side a + b + c has a 3 × 3 grid of 9 pieces.
• (a + b)³ = a³ + 3a²b + 3ab² + b³: a cube of side a + b splits into 8 blocks.
• (a − b)³ = a³ − 3a²b + 3ab² − b³: put −b in place of b.
• a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²).
• If a + b + c = 0 then a³ + b³ + c³ = 3abc.

Always check a new identity with numbers, for example a = 2, b = 1.

Simplifying rational expressions

A rational expression is one polynomial divided by another, like (x² − 9)/(x + 3). To simplify, factorise the top and the bottom, then cancel common factors.

(x² − 9)/(x + 3) = (x + 3)(x − 3)/(x + 3) = x − 3, for x ≠ −3.

Remember: cancel only whole factors, never single terms. And the value that makes the bottom zero is not allowed.

Try it: a practical

Cut paper: one 7 cm square (a²), two 7 cm × 3 cm strips (ab) and one 3 cm square (b²). Join them into one big square. Measure its side: 10 cm. Check 10² = 49 + 42 + 9 = 100. Then cut 12 unit squares and strips to factorise x² + 7x + 12 yourself.

Board exam corner

Typical questions: expand using an identity; evaluate 99³ or 104 × 96 without multiplying directly; factorise x³ + 27 or 8a³ − 27b³; use a + b + c = 0; prove an identity with a figure; simplify a rational expression.

Key formulas and definitions

• (a + b)² = a² + 2ab + b²
• (a − b)² = a² − 2ab + b²
• a² − b² = (a + b)(a − b)
• (x + a)(x + b) = x² + (a + b)x + ab
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
• (a + b)³ = a³ + b³ + 3ab(a + b); (a − b)³ = a³ − b³ − 3ab(a − b)
• a³ + b³ = (a + b)(a² − ab + b²); a³ − b³ = (a − b)(a² + ab + b²)
• a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

Worked examples

Common mistakes

• Writing (a + b)² = a² + b². The two ab strips are missing: the middle term 2ab is needed.
• Getting the sign wrong in (a − b)²: the last term + b² is always positive.
• Cancelling terms in fractions: (x + 3)/3 is not x. Cancel only common factors.
• Forgetting to square the coefficient: (3x)² = 9x², not 3x².