Exploring Algebraic Identities

Class 09 Maths

Algebraic identities are special mathematical rules that not only make it easier to simplify complicated calculations but also help us work efficiently with algebraic expressions.

Identities are equations that are true for all values of the variables.

Formula for Identities

$$ (x + y)^2 = x^2 + 2xy + y^2 $$

$$ (x - y)^2 = x^2 - 2xy + y^2 $$

$$ (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx $$

$$ (x + y)(x - y) = x^2 - y^2 $$

$$ (x + a)(x + b) = x^2 + (a + b)x + ab $$

$$ (ax + b)(cx + d) = acx^2 + (ad + bc)x + bd $$

$$ x^3 - y^3 = (x - y)(x^2 + xy + y^2) $$

$$ x^3 + y^3 = (x + y)(x^2 - xy + y^2) $$

$$ (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 $$

$$ (x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3 $$

$$ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz) $$

Examples

Example 1: The sum of three numbers is 10 and their product is 25. The sum of their squares is 38. Find the sum of the cubes of these three numbers.

Let the three numbers be x, y and z, respectively. According to the problem

x + y + z = 10

xyz = 25

x² + y² + z² = 38

Substituting in the identity

(x + y + z)(x² + y² + z² – xy – xz – yz) = x³ + y³ + z³ – 3xyz, we get

(10)(38 – xy – xz – yz) = x³ + y³ + z³ – 3(25)

x³ + y³ + z³ = 380 – 10(xy + xz + yz) + 75 = 455 – 10(xy + xz + yz)

To find (xy + xz + yz), use the identity

(x + y + z)² = x² + y² + z² + 2xy + 2xz + 2yz

100 = 38 + 2(xy + xz + yz)

xy + xz + yz = 31

Now, substituting this into earlier equation,

x³ + y³ + z³ = 455 – 10(xy + xz + yz) = 455 – 10(31) = 455 – 310 = 145

Example 2: A rectangular pool is such that its breadth is 4 metres less than its length and its area is 96 sq. metres. Find the length and breadth of the pool.

Let the length of the pool be x metres. Then the breadth of the pool is x – 4 metres.

Since the area is given to be 96 sq. metres, we get

x(x – 4) = 96

x² – 4x = 96

x² – 4x – 96 = 0

x² – 12x + 8x – 96 = 0

x(x – 12) + 8(x – 12) = 0

(x – 12)(x + 8) = 0

This means either x – 12 = 0 or x + 8 = 0

x = 12 or x = -8

Since x is the length of the pool, it cannot be negative. Therefore, we ignore x = – 8 and x = 12 metres must be the length of the pool. The breadth of the pool = x – 4 = 12 – 4 = 8 metres.