hw7 - (1) (a) Let f (x), g(x) be polynomials with real...

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Unformatted text preview: (1) (a) Let f (x), g(x) be polynomials with real coefficients. Prove that there exists a unique polynomial h(x) = xk + ak-1 xk-1 + . . . + a0 such that h(x) divides both f (x) and g(x) and every other polynomial that divides both f (x) and g(x) divides h(x). h(x) is called the greatest common divisor of f (x) and g(x) and is denoted by (f (x), g(x)). Hint: Use Euclidean algorithm to construct h(x). (b) Find (x3 + x2 - 5x + 3, x4 + 3x3 - x - 3). (2) Find all rational solutions of x3 - x2 - x - 2 = 0. (3) Let z0 be a root of xn = z. Show that all roots of xn - z = 0 have the form z0 k where 0 , . . . n-1 are n-th roots of 1. 1 ...
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