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

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Question

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

Options:

A . x5+72x4+140x3−580x2+1024x+720=0

B . x5−20x4+155x3−580x2+1044x−720=0

C . x5+20x4+165x3+585x2−1044x−720=0

D . x5−30x4−165x3−680x62−1024x+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

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