Is the polynomial always divisible by a given number?

Algebra Level 2

You are given a polynomial P(x) = x 2 + p x + q x^{2}+px+q . How should p and q be so that P(x) always even with all integer x?

p is even meanwhile q is odd. p and q are both odd. p is odd meanwhile q is even. p and q are both even. P(x) is always odd, so there are no p and q.

Tin Le by Tin Le , 15, Vietnam

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2 solutions

Jordan Cahn
Apr 9, 2019

If x x is even then x 2 + p x x^2+px will be even. Thus, for P ( x ) P(x) to be even, q q must be even.

Now, if x x is odd, x 2 + q x^2+q will be an odd plus an even and, therefore, odd. Thus p x px and, by extention, p p , must be odd.

2 Helpful 1 Interesting 0 Brilliant 0 Confused

For x=2m, where m is an integer, P(x)=4m^2+2mp+q is even. This means q is even. For x=2m+1, P(x)=(2m+1)^2+(2m+1)p+q is even with even q. This means p is odd.

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