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Let P(x) = x2 + ax + b be a quadratic polynomial where a is real and b ≠ 2 is rational. Suppose P(0)2, P(1)2, P(2)2 are integers. Prove that a and b are integers. |
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Answer» We have P(0) = b. Since b is rational and b2 = P(0)2 is an integer, we conclude that b is an integer. Observe that P(1)2 = (1 + a + b)2 = a2 + 2a(1 + b) + (1 + b)2 ∈ Z P(2)2 = (4 + 2a + b)2 = 4a2 + 4a(4 + b) + (4 + b)2 ∈ Z Eliminating a2, we see that 4a(b - 2) + 4(1 + b)2 - (4 + b)2 ∈ Z. Since b ≠ 2, it follows that a is rational. Hence the equation x2 + 2x(1 + b) + (1 + b)2 - (a2 + 2a(1 + b) + (1 + b)2) = 0 is a quadratic equation with integer coeffcients and has rational solution a. It follows that a is an integer. |
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