Let a > 0, and let P (x) be a polynomial with integer coefficients such that P (

Question

Let a > 0, and let P (x) be a polynomial with integer coefficients such that P (l) = P (3) = P(5) = P (7) = a, and P (2) = P (4) = P (6) = P (8) = -a. What is the smallest possible value of a? (A) 105 (B) 315 (C) 945 (D) 7! (E) 8! Monic quadratic polynomials P (x) and Q (x) have the property that P (Q(x)) has zeros at x = -23, -21, -17, and -15, and Q (P (x)) has zeros at x = -59, -57, -51 and -49. What is the sum of the minimum values of P (x) and Q (x)? (A) -100 (B) -82 (C) -73 (D) -64 (E) 0

Explanation / Answer

Ans(1):

Given that P(1) = P(3) = P(5) = P(7) = a
lets define polynomial R(x) such that

R(x) = P(x) - a...(i)

Then x=1,3,5,7 will be roots of R(x)
As x=1,3,5,7 are roots of R(x), so we can write R(x) as

R(x)=(x-1)(x-3)(x-5)(x-7)(Q(x))...(ii)

using (i) and (ii) we get:
P(x) - a=(x-1)(x-3)(x-5)(x-7)(Q(x))...(iii)
plugx=2,4,6,8 respectively we get:

P(2)-a=(2-1)(2-3)(2-5)(2-7)Q(2)
or -a-a = -15Q(2) {given P(2)=-a}
or -2a = -15Q(2)...(iv)

similarly
P(4)-a=(4-1)(4-3)(4-5)(4-7)Q(4)
or -2a = 9Q(4)...(v)

similarly
P(6)-a=(6-1)(6-3)(6-5)(6-7)Q(6)
or -2a = -15Q(6)...(vi)

similarly
P(8)-a=(8-1)(8-3)(8-5)(8-7)Q(8)
or -2a = 105Q(8)...(vii)

Using (iv),(v),(vi),(vii) we get:
-2a=-15Q(2)=9Q(4)=-15Q(6)=105Q(8)
So the least value of a must be the LCM of {15,9,15,105} which is 315.

Hence correct answer is B) 315.

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