Short cut cubing of any three digit number just in few seconds with help of Horner`s method for polynomial evaluation

We can utilize following algebraic expansion of Horner`s method for polynomial evaluation to easily find cube of any three digit number

(z + d) ³ = z x [z x (z + 3d) + 3d²] + d³

Where, d will always be one of the numbers between (+/-) 1 to (+/-) 50.

For example,

To find cube of the number 323,

Let, z = 300 and d = 23

So,

(z + d) ³ = z x [z x (z + 3d) + 3d²] + d³

(300 + (23)) ³ = 300 x [300 x (300 + 3(23)) + 3(23)²] + (23)³

(323)³ = 300 x [300 x (300 + 69) + 3(23)²] + (23)³

= 300 x [300 x 369 + 3(23)²] + (23)³

= 300 x [300 x 369 + 1587] + (23)³

= 300 x [110700 + 1587] + (23)³

= 300 x 112287 + (23)³

= 33686100 + 12167

= 33698267

Short cut cubing of any two digit number just in few seconds with help of Horner`s method for polynomial evaluation

We can utilize following algebraic expansion of Horner`s method for polynomial evaluation to easily find cube of any two digit number

(z + d) ³ = z x [z x (z + 3d) + 3d²] + d³

Where, d will always be one of the numbers (+/-)1, (+/-)2, (+/-)3, (+/-)4, (+/-)5.

So, 3d2 will always be one of the numbers 3, 12, 27, 48, or 75.

For example,

To find cube of the number 23,

Let, z = 20 and d = 3

So,

(z + d) ³ = z x [z x (z + 3d) + 3d²] + d³

(20 + 3)³ = 20 x [20 x (20 + 3(3)) + 3(3)²] + (3)³

(23)³ = 20 x [20 x (20 + 9) + 27] + (3)³

= 20 x [20 x 29 + 27] + (3)³

= 20 x [580 + 27] + (3)³

= 20 x 607 + (3)³

= 12140 + 27

= 12167

Another example,

To find cube of the number 88,

Let, z = 90 and d = (-2)

So,

(z + d) ³ = z x [z x (z + 3d) + 3d²] + d³

(90 + (-2)) ³ = 90 x [90 x (90 + 3(-2)) + 3(-2)²] + (-2)³

(88)³ = 90 x [90 x (90 + (-6)) + 12] + (-2)³

= 90 x [90 x 84 + 12] + (-2)³

= 90 x [7560 + 12] + (-2)³

= 90 x 7572 + (-2)³

= 681480 + (-8)

= 681472