The number of polynomials p(x) with integer coefficients such that the

Question

The number of polynomials p(x) with integer coefficients such that the curve y=p(x) passes through (6, 8) and (4, 9) is

A. 0
B. 1
C. 2
D. 3
E. infinite

nick1816 · Answer

x and y both are even at (6,8). Now if we replace value of x with another even number, y must be even for that value of x too, as coefficient's are integer. But at x=4, we are getting odd value of y. No such polynomial is possible.

skmaqeel51 · Answer

why do both have to be even?

egmat · Answer

Hi skmaqeel51,

Good question, because nick's line is a little compressed and it's easy to read it as "even x always gives even y." That's  not  the rule. A polynomial like p(x) = x + 1 gives an odd y at x =  6 . So let's pin down what's actually being claimed.

The real constraint is about the  difference  between the two y-values.

With integer coefficients, write p(x) as a sum of terms c·x^k. Look at p( 6 ) - p( 4 ):

- Each term contributes c·( 6 ^k -  4 ^k).
- Every  6 ^k -  4 ^k is divisible by  6  -  4  =  2  (even minus even, and the pattern holds for any power).
- So the whole thing, p( 6 ) - p( 4 ), must be  even .

An even difference means p( 6 ) and p( 4 ) have the  same parity  - either both even or both odd. That's the actual fact nick was leaning on.

Now apply it: the point ( 6 ,  8 ) forces p( 6 ) =  8 , which is  even . Same parity then forces p( 4 ) to also be  even . But the other point demands p( 4 ) =  9 , which is  odd . Impossible - so  no such polynomial exists , and the answer is  A .

Quick checks to feel the rule:

- p(x) = x + 1: p( 4 ) =  5 , p( 6 ) =  7  - both odd (same parity).
- p(x) = x^2: p( 4 ) =  16 , p( 6 ) =  36  - both even (same parity).

Every integer-coefficient polynomial does this: at two even inputs, the two outputs always match in parity. You can never get one even ( 8 ) and one odd ( 9 ) - which is exactly why the problem breaks.

Answer: A

The number of polynomials p(x) with integer coefficients such that the

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The number of polynomials p(x) with integer coefficients such that the

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