Find the equation with roots 2, 3, 4, 5 and 6. Options: A . &nbsp x 5 + 72 x 4 + 140 x 3 − 580 x 2 + 1024 x + 720 = 0 B . &nbsp x 5 − 20 x 4 + 155 x 3 − 580 x 2 + 1044 x − 720 = 0 C . &nbsp x 5 + 20 x 4 + 165 x 3 + 585 x 2 − 1044 x − 720 = 0 D . &nbsp x 5 − 30 x 4 − 165 x 3 − 680 x 62 − 1024 x + 720 = 0

Answer: Option B : B Let the equation be A x 5 + B x 4 + C x 3 + D x 4 + E x + F = 0 Sum of the product of the roots taken 1 at a time = − B A → = − B A Sum of the product of the roots taken 2 at a time = C A → ( 2 × 3 ) + ( 2 × 4 ) + ( 2 × 5 ) + ( 2 × 6 ) + ( 3 × 4 ) + ( 3 × 5 ) + ( 3 × 6 ) + ( 4 × 5 ) + ( 4 × 6 ) + ( 5 × 6 ) = C A → 155 = C A Sum of the product of the roots taken 3 at a time = − D A → ( 2 × 3 × 4 ) + ( 2 × 3 × 5 ) + ( 2 × 3 × 6 ) + ( 2 × 4 × 5 ) + ( 2 × 4 × 6 ) + ( 2 × 5 × 6 ) + ( 3 × 4 × 5 ) + ( 3 × 4 × 6 ) + ( 3 × 5 × 6 ) + ( 4 × 5 × 6 ) = − D A → 580 = − D A Sum of the product of the roots taken 4 at a time = E A → ( 2 × 3 × 4 × 5 ) + ( 2 × 3 × 4 × 6 ) + ( 2 × 3 × 5 × 6 ) + ( 2 × 4 × 5 × 6 ) + ( 3 × 4 × 5 × 6 ) = 1044 → 1044 = E A Sum of the product of the roots taken 5 at a time = − F A → ( 2 × 3 × 4 × 5 × 6 ) = − F A → 720 = − F A Take A=1. Then B= -20,C=155, D= -580, E=1044, F=-720

Find the equation with roots 2, 3, 4, 5 and 6.

Question

Find the equation with roots 2, 3, 4, 5 and 6.

Options:

A .

x^{5} + 72 x^{4} + 140 x^{3} - 580 x^{2} + 1024 x + 720 = 0

B .

x^{5} - 20 x^{4} + 155 x^{3} - 580 x^{2} + 1044 x - 720 = 0

C .

x^{5} + 20 x^{4} + 165 x^{3} + 585 x^{2} - 1044 x - 720 = 0

D .

x^{5} - 30 x^{4} - 165 x^{3} - 680 x 62 - 1024 x + 720 = 0

Answer: Option B
:
B

Let the equation be $A x^{5} + B x^{4} + C x^{3} + D x^{4} + E x + F = 0$
Sum of the product of the roots taken 1 at a time $= \frac{- B}{A}$
$\to= - \frac{B}{A}$
Sum of the product of the roots taken 2 at a time $= \frac{C}{A}$
$\to (2 \times 3) + (2 \times 4) + (2 \times 5) + (2 \times 6) + (3 \times 4) + (3 \times 5) + (3 \times 6) + (4 \times 5) + (4 \times 6) + (5 \times 6) = \frac{C}{A}$
$\to 155 = \frac{C}{A}$
Sum of the product of the roots taken 3 at a time $= - \frac{D}{A}$
$\to (2 \times 3 \times 4) + (2 \times 3 \times 5) + (2 \times 3 \times 6) + (2 \times 4 \times 5) + (2 \times 4 \times 6) + (2 \times 5 \times 6) + (3 \times 4 \times 5) + (3 \times 4 \times 6) + (3 \times 5 \times 6) + (4 \times 5 \times 6) = - \frac{D}{A}$
$\to 580 = - \frac{D}{A}$
Sum of the product of the roots taken 4 at a time $= \frac{E}{A}$
$\to (2 \times 3 \times 4 \times 5) + (2 \times 3 \times 4 \times 6) + (2 \times 3 \times 5 \times 6) + (2 \times 4 \times 5 \times 6) + (3 \times 4 \times 5 \times 6) = 1044$
$\to 1044 = \frac{E}{A}$
Sum of the product of the roots taken 5 at a time $= - \frac{F}{A}$
$\to (2 \times 3 \times 4 \times 5 \times 6) = - \frac{F}{A}$
$\to 720 = - \frac{F}{A}$
Take A=1. Then B= -20,C=155, D= -580, E=1044, F=-720