Question

Let p(x) be a polynomial of degree 4 such that p(1) = p(3) =p(5) =p(7)=0. if the real number x not equal to 1,3,5 is such that p(x) = 0 can be expressed as x = p/q where p and q are relatively prime, then (p-8q) is ?

Answer 1

Answer:

ʜᴇʀᴇ ɪs ᴜʀ ᴀɴsᴡᴇʀ ʜᴏᴘᴇ ɪᴛ ᴡɪʟʟ ʜᴇʟᴘ ᴜ

Answer 2

Let p(x) = a(x-1)(x-3)(x - 5)( x - α)

Taking log on both sides,

log p(x) = log a  + log(x-1) + log(x-3) + log(x-5) +log(x-7)

Differentiating both sides wrt. x,

1/p(x) = 1/(x-1) + 1/(x-3) + 1/(x-5) + (1/x-α)

Since, Derivative of p(7) = 0,

0 = 1/6 + 1/4 +1/2 + 1/(7-α)

α = 89/11 = p/q

So, p - 8q = 1