ALGEBRAIC IDENTITIES

ALGEBRAIC IDENTITIES

Let us recall the following algebraic identities is earlier.

(a + b)² = a² + 2ab + b^2.                                    (a – b)² = a² – 2ab + b^2.                                  (a + b) (a – b) = a² – b^2.                                    (x + a) (x + b) = x² + (a + b) x + ab

Ex. Expand each of the following :

(i) (2x + 3y)²

(ii) (4x – 5y)²

(iii) (x + 5) (x + 6)

(iv) (x – 3) (x – 5)

Sol. (i) (2x + 3y)^2.=                               (2x)² + 2.2x.3y + (3y)²                             b   [∵ (a + b)² = a² + 2ab + b²]

= 4x² + 12xy + 9y²

(ii) (4x – 5y)² = (4x)² + 2.4x.5y + (5y)2                        [∵ (a – b)² = a²– 2ab + b²]

= 16x² – 40xy + 25y²

(iii) (x + 5) (x + 6) = x² + (5 + 6)x + 5.6               [∵ (x + a) (x + b) = x² + (a + b) x + ab]

= x² + 11x + 30

(iv) (x – 3) (x – 5) = {x + (–3)} {x + (–5)}            [∵ (x + a) (x + b) = x² + (a + b) x + ab]

= x² + {(–3) + (–5)}x + (–3).(–5) = x² + (–3 – 5)x + 15.                                                  = x² – 8x + 15

Ex. Find the product using appropriate identities :

(i) (x + 8) (x + 8)

(ii) (3x – 2y) (3x – 2y)

(iii) (x + 0.1) (x – 0.1)

Sol. (i) (x + 8) (x + 8) = (x + 8)^2.                       = x²+ 2(x) × 8 + (8)² = x² + 16x + 64.

(ii) (3x – 2y) (3x – 2y) = (3x – 2y)²                          [∵ (a – b)² = a² – 2ab + b²]

= (3x)² – 2(3x) × 2y + (2y)^2.                             = 9x² – 12xy + 4y².

(iii) (x + 0.1) (x – 0.1) = (x)2 – (0.1)²                       [∵ (a + b) (a – b) = a² – b²]

= (x)² – 0.01