By definition, p(x) is a polynomial with integer coefficients if p(x) = ao + a,x +...

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By definition, p(x) is a polynomial with integer coefficients if p(x) = ao + a,x + a2x2 + ..+ a,nx" for some non-negative integer n, integers a, a1, az, ., an and variable x. The number r is a root of the polynomial p(x) if p(r) = 0. .... (a) Suppose a and b are integers and p(x) is a polynomial with integer coefficients. Prove that if a + bv2 is a root of p(x), then a – by2 is also a root of p(x). (b) Find a polynomial q(x) with integer coefficients, and integers c and d such that c+dV2 is a root of q(x), but c - dV2 is not a root of q (x), where V2 denotes the cube root of 2. Describe conditions on c, d, and q(x) for this to hold. By definition, p(x) is a polynomial with integer coefficients if p(x) = ao + a,x + a2x2 + ..+ a,nx" for some non-negative integer n, integers a, a1, az, ., an and variable x. The number r is a root of the polynomial p(x) if p(r) = 0. .... (a) Suppose a and b are integers and p(x) is a polynomial with integer coefficients. Prove that if a + bv2 is a root of p(x), then a – by2 is also a root of p(x). (b) Find a polynomial q(x) with integer coefficients, and integers c and d such that c+dV2 is a root of q(x), but c - dV2 is not a root of q (x), where V2 denotes the cube root of 2. Describe conditions on c, d, and q(x) for this to hold.