A regular n-gon is constructible if and only if cos(2π/n) is constructible. (a) Prove that a regular hexagon is constructible. (b) Prove that a regular 7-gon is not constructible because cos(2π/7) is a root of the irreducible polynomial 8x³ + 4x² - 4x - 1.

1) Consider the polynomial f(x) = x4+1(a) Explain why f has no real roots, and why this means f must factor as aproduct of two irreducible quadratics.(b) Factor f and find all of its complex roots.   2.) Let f be the polynomial function of degree 4 with real coefficients, leadingcoefficient 1, and roots x = 2 + i, 3, −3. Let g be the polynomial function of degree 4 withintercept (0, −3) and roots x = i, 3i. Find (f + g)(1).

Let p(x) = x3 + x + 10 a) Show that x = 1+2i is a root of p(x) where i = surds(-1). b) By using the fact that p(x) is a polynomial with real coefficients, show by contradiction that  p(x) will have another complex root.

For z=1+i find the product ,, of z and its complex conjugate zZ

Let n eN (that is, n is a positive whole number). The n-th root of a complex number z is z1/n. When dealing with roots we have to be careful since the n-th root function is multi-valued. In particular, there are n different possible values of w such that w" = z, all of which could properly be considered n-th roots of z! Let's explore this! ein/2. Show that eir/6, e5in/6, and e3it/2 (g) Consider the cube roots of i of i. Plot these roots on the complex plane and demonstrate graphically how ešin/6 cubes to i. [Note: Remember that multiplying by a phase e?º essentially "rotates" a point in the complex plane counterclockwise by an angle 0.] are all cube roots Find a formula for the n distinct values of the n-th root of z = rei®. As a check, test that your (h) formula implies that if w is a square root of z then so is -w.