Purana Apurnabhyam

### Sanskrit Name:

पूरणापूरणाभ्यां

### English Translation:

By the completion and non completion (of square, cube, fourth power, etc).

### Prerequisites:

It uses the technique of completion of polynomials with Paravartya sutra to find its factors.

Examples:

1) x^{3} + 9x^{2} + 24x + 16 = 0 i.e. x^{3} + 9x^{2} = -24x -16

We know that (x+3)^{3} = x^{3}+9x^{2}+27x+27 = 3x + 11 (Substituting above step).

i.e. (x+3)^{3} = 3(x+3) + 2 … (write 3x+11 in terms of LHS so that we substitute a term by a single variable).

Put y = x+3

So, y^{3} = 3y + 2

i.e. y^{3} – 3y – 2 = 0

Solving using the methods discussed (coeff of odd power = coeff of even power) before.

We get (y+1)^{2} (y-2) = 0

So, y = -1 , 2

Hence**x = -4,-1**

2) x^{3} + 7x^{2} + 14x + 8 = 0 i.e. x^{3} + 7x^{2} = – 14x – 8

We know that (x+3)^{3} = x^{3}+9x^{2}+27x+27 = 2x^{2} + 13x + 19 (Substituting above step).

i.e. (x+3)^{3}= 2x^{2} + 13x + 19

Now we need factorize RHS in terms of (x+3). So apply Paravartya sutra.

So Dividing 2x^{2} + 13x + 19 by (x+3) gives

2x^{2} + 13x + 19 = (x+3)(2x-7)-2

i.e. (x+3)^{3} = (x+3)(2x-7)-2

put y = x+3

So, y^{3} = y(2y+1) -2

Solving gives y = 1,-1,2

Hence**x= -2, 4, 1, -1**

3) x^{4} + 4x^{3} – 25x^{2} – 16x + 84 = 0i.e. x^{4} + 4x^{3} = 25x^{2} + 16x – 84

We know that (x+1)^{4} = X^{4} + 4x^{3} + 6x^{2} + 4x +1 = 31×2 + 20x – 83 (Substituting above step).

i.e. (x+1)^{4} =31x^{2} + 20x – 83

Now we need factorize RHS in terms of (x+1). So apply Paravartya sutra.

So Dividing 31x^{2} + 20x – 83 by (x+1) gives

31x^{2} + 20x – 83 = (x+1)(31x-11) – 72

i.e. (x+1)^{4} = (x+1)(31x-11) – 72

put y= x+1

So, y^{4} = y(31y-42) – 72

i.e. y^{4} – 31y^{2} + 42y + 72 = 0

On Factorization (Sum of coeff of even power = coeff of odd power. So (y+1) is 1 factor).

Dividing above biquadratic eq by (y+1) will give cubic eq, Further factorization gives

y = -1,3,4 or -6

Hence **x = -2, 2 ,3 or -7**

Vyasti Samasti Sutra

### Sanskrit Name:

व्यष्टि समष्टि

### English Translation:

Individuality or Totality(Vyasti: Individual; Samasti: Totality)

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