Solving algebraic problems get easy with readily available expressions. These expressions are knows as algebraic identities or rules which makes the job of a person easy, fast and reduces the chances of error. Below are most commonly used algebraic identities:
Square of a Binomial
(a + b)2 = a2 + 2ab + b2
Example: (x + 4)2 = x2 + 2 × x × 4 + 42 = x2 + 8 x + 16
(a – b)2 = a2 – 2ab + b2
Example: (x – 5)2 = x2 – 2 × x × 5 + 52 = x2 + 10 x + 25
Difference of Squares
(a + b)(a – b) = a2 – b2
(x + 3)(x – 3) = x2 – 32 = x2 – 9
Cube of a Binomial
(a + b)3 = a3 + 3a2b + 3ab2 + b3
Example: (x + 2)3 = x3 + 3 × x2 × 2 + 3 × x × 22 + 23 = x3 + 6x2 + 12x + 8
(a – b)3 = a3 – 3a2b + 3ab2 – b3
Example: (x – 4)3 = x3 – 3 × x2 × 4 + 3 × x × 42 – 43 = x3 – 12x2 + 48x – 64
Sum of Cubes
a3 + b3 = (a + b)(a2 – ab + b2)
Example: x3 + 8 = x3 + 23 = (x + 2)(x2 – 2x + 22) = (x + 2)(x2 – 2x + 4)
Difference of cubes
a3 – b3 = (a – b)(a2 + ab + b2)
x3 – 27 = x3 – 33 = (x – 3)(x2 + 3x + 32) = (x – 3)(x2 + 3x + 9)
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