Let f(x)= px^5 + qx^4 + rx^3 + sx^2 + tx + u be a polynomial such that f(1) = -1 and f(2) = 3 and all the numbers p, q, r, s, t and u are integers. prove that the equation
px^5 + qx^4 + rx^3 + sx^2 + tx + u = 0
has no integer roots.
To prove that the equation has no integer roots, we can use the method of contradiction.
Assume that the equation px^5 + qx^4 + rx^3 + sx^2 + tx + u = 0 has an integer root, let's call it a.
Substituting a into the equation, we have:
pa^5 + qa^4 + ra^3 + sa^2 + ta + u = 0
Since a is an integer, all the terms on the left side of the equation are also integers.
Now, let's consider the values of f(1) and f(2) given in the problem. By substituting x = 1 and x = 2 into the equation f(x), we get:
f(1) = p(1)^5 + q(1)^4 + r(1)^3 + s(1)^2 + t(1) + u = p + q + r + s + t + u
f(2) = p(2)^5 + q(2)^4 + r(2)^3 + s(2)^2 + t(2) + u = 32p + 16q + 8r + 4s + 2t + u
Given that f(1) = -1 and f(2) = 3, we can form two equations:
p + q + r + s + t + u = -1 (Equation 1)
32p + 16q + 8r + 4s + 2t + u = 3 (Equation 2)
Now, let's subtract Equation 1 from Equation 2:
31p + 15q + 7r + 3s + t = 4 (Equation 3)
From Equation 3, we can observe that the left side of the equation contains a linear combination of 31p, 15q, 7r, 3s, and t. Since the coefficients are all integers and the right side is an integer, the left side must also be an integer.
Now, let's consider the equation pa^5 + qa^4 + ra^3 + sa^2 + ta + u = 0, which we assumed has an integer root a.
Substituting the values of p, q, r, s, and t into Equation 3, we can rewrite it as:
31p(a^4) + 15q(a^4) + 7r(a^3) + 3s(a^2) + t(a) = -u
Since all terms on the left side of the equation are integers, the right side (-u) must also be an integer. This implies that u is an integer.
However, if we go back to Equation 1, we see that u = -1 - (p + q + r + s + t). Since the sum of integers is an integer, the right side (-1 - (p + q + r + s + t)) is an integer.
Thus, we have reached a contradiction because u cannot be both an integer and not an integer. Therefore, our assumption that the equation px^5 + qx^4 + rx^3 + sx^2 + tx + u = 0 has an integer root is false.
Hence, the equation has no integer roots.