Exploring Algebraic Identities Class 9: Notes and Solved Examples

Exploring Algebraic Identities is chapter 4 of the new Class 9 Mathematics textbook. In this guide you will learn what an identity is, how each identity in the chapter works (with its geometric picture), how to use identities for quick calculations and factorising, and how to simplify rational expressions. Solved important questions and common mistakes come at the end.

Identity or equation?

An algebraic identity is true for every value of its variables. An equation is true only for some values.

• x² - 1 = 24 is an equation. It is true only when x = 5 or x = -5.
• (x + y)² = x² + 2xy + y² is an identity. It works for any x and y.

Identities are shortcuts. They let you expand products or factorise expressions without long multiplication.

A neat pattern: take three consecutive numbers with middle number n. Their squares are (n - 1)², n² and (n + 1)². Then: (n - 1)² + (n + 1)² - 2n² = (n² - 2n + 1) + (n² + 2n + 1) - 2n² = 2. The answer is always 2, whatever n is.

Square of a sum: (a + b)² = a² + 2ab + b²

Picture it: draw a square of side (a + b). Split it into one square of area a², one square of area b² and two rectangles of area ab each. Total area: a² + 2ab + b².

It works for negative numbers too. With a = -2 and b = -3: (a + b)² = (-5)² = 25, and a² + 2ab + b² = 4 + 12 + 9 = 25.

Example: (5x + 2y)² = (5x)² + 2(5x)(2y) + (2y)² = 25x² + 20xy + 4y².

Quick squaring: 43² = (40 + 3)² = 1600 + 240 + 9 = 1849.

Square of a difference: (a - b)² = a² - 2ab + b²

Replace b with -b in the first identity and you get this one.

Picture it: start with a square of side a (area a²). Remove a rectangle of area ab and another of area b(a - b). What is left is the small square of side (a - b).

This is handy for numbers just below a round number.

Example: 29² = (30 - 1)² = 900 - 60 + 1 = 841. Example: (2x - 3y)² = 4x² - 12xy + 9y².

Square of three terms

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

To see why, call (b + c) = d. Then (a + d)² = a² + 2ad + d². Put (b + c) back for d and expand (b + c)².

Example: 119² = (100 + 10 + 9)² = 10000 + 100 + 81 + 2000 + 1800 + 180 = 14161.

The identity also finds a sum of squares. If a + b + c = 10 and ab + bc + ca = 31, then 100 = a² + b² + c² + 62, so a² + b² + c² = 38.

Difference of squares: a² - b² = (a + b)(a - b)

This is used both to expand and to factorise. It can also be written as a² = (a + b)(a - b) + b². In 750 CE, the mathematician Śhrīdharāchārya proposed this as a way to compute squares quickly.

Example: 55² = (55 + 5)(55 - 5) + 5² = 60 × 50 + 25 = 3025. Factorise: 49x² - 16y² = (7x)² - (4y)² = (7x + 4y)(7x - 4y).

Factorising quadratics: x² + (a + b)x + ab = (x + a)(x + b)

This is splitting the middle term. Find two numbers that add to the coefficient of x and multiply to the constant.

• x² + 7x + 12: numbers 3 and 4 (3 + 4 = 7, 3 × 4 = 12), so (x + 3)(x + 4). With algebra tiles, one x² tile, seven x-tiles and twelve unit tiles make a rectangle with sides (x + 3) and (x + 4).
• x² - 5x + 6: numbers -2 and -3, so (x - 2)(x - 3).
• If the constant is negative, the two numbers have different signs.

Cubic identities

• (a + b)³ = a³ + 3a²b + 3ab² + b³. A cube of side (a + b) splits into two cubes (a³ and b³) and six cuboids (three of volume a²b and three of volume ab²).
• (a - b)³ = a³ - 3a²b + 3ab² - b³. The signs alternate.

Example: (x - 2y)³ = x³ - 6x²y + 12xy² - 8y³.

Example: a cube has volume p³ + 6p²q + 12pq² + 8q³. Rewrite it as p³ + 3(p²)(2q) + 3(p)(2q)² + (2q)³. This is (a + b)³ with a = p and b = 2q, so the side is (p + 2q).

Sum and difference of cubes

• x³ + y³ = (x + y)(x² - xy + y²)
• x³ - y³ = (x - y)(x² + xy + y²)
• x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)

From the last one: if x + y + z = 0, then x³ + y³ + z³ = 3xyz, because the right side becomes zero.

Simplifying rational expressions

A rational expression is a fraction whose top and bottom are polynomials. Factorise both, then cancel common factors (the denominator must not be zero).

Example: (x² - 7x + 12) / (5x² + 5x - 100) Top: (x - 3)(x - 4). Bottom: 5(x - 4)(x + 5). Cancel (x - 4) to get (x - 3) / 5(x + 5).

Remember this

Important questions with answers

1. How is an identity different from an equation? An identity is true for all values of its variables; an equation is true only for particular values.

2. Is (a + b)² equal to a² + b²? No. (a + b)² = a² + 2ab + b². It equals a² + b² only when 2ab = 0, that is, when a or b is zero.

3. Find 102² using an identity. (100 + 2)² = 10000 + 400 + 4 = 10404.

4. Find 98² using an identity. (100 - 2)² = 10000 - 400 + 4 = 9604.

5. Find 101 × 99 quickly. (100 + 1)(100 - 1) = 100² - 1² = 10000 - 1 = 9999.

6. Factorise x² + 11x + 30. 5 + 6 = 11 and 5 × 6 = 30, so (x + 5)(x + 6).

7. Simplify (x² - 9) / (x + 3). x² - 9 = (x + 3)(x - 3). Cancel (x + 3) to get x - 3.

8. If x + y + z = 0, what is x³ + y³ + z³? It equals 3xyz.

9. Expand (3a + b)². (3a)² + 2(3a)(b) + b² = 9a² + 6ab + b².

Common mistakes to avoid

• Writing (a + b)² = a² + b². The middle term 2ab is missing.
• Forgetting the sign in (a - b)³: it is a³ - 3a²b + 3ab² - b³.
• Squaring only the letter: (5x)² is 25x², not 5x².
• Mixing up x³ - y³ = (x - y)(x² + xy + y²) with the sum of cubes, where the middle sign is minus.
• Cancelling terms instead of factors in rational expressions. Factorise first, then cancel.

To practise these identities with step-by-step help, study this chapter with Joy on Learnijoy.