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MATH1050 Additional practice problems - Section 5 1. Simplify the following expressions. √ √ √ (a) ( 8 − 18)2 (b) 20 √ 2− 5 2. Solve the following inequalities. Illustrate your solutions on the real number line and state your answers in interval format. 2 (a) 2x − 1 < 5x + 7 (b) x2 ≥ 2x + 8 (c) ≤5 x−3 3. Solve the following equations. (a) |3 − 4x| = 15 (b) 1 − y = |y − 2| 2x − 1 =3 (c) x+1 4. Solve the following inequalities. Illustrate your solutions on the real number line and state your answers in interval format. 2 − x |x − 5| ≤3 (a) |3 − 2x| < 4 (b) (c) 3 < x+1 x+2 5. Let u = 4i, w = 2 − 3i and z = 1 + 5i. Write the following in Cartesian form. u (a) 3w − z (b) wz (c) u (d) z 6. Let z = a + bi and w = c + di be complex numbers. Prove the following. (a) |z| = |z| (b) z + w = z + w √ 7. Let z = − 3 + i and let w = 2 − 2i. (a) Write each of z and w in polar form. (b) Use the polar forms of z and w to determine zw and w . z (c) Use De Moivre’s Theorem to determine z 5 and w6 (in polar form). π 8. Let z = 6e− 3 i . Write z in Cartesian form. 9. (a) Determine the cube roots of 1 + i. (b) Suppose w6 = −64i. Determine the six values of w. 10. Determine which of (z − 2), (z + 3) and (z + 4) is a factor of 2z 3 + 9z 2 − 6z − 40. Hence write 2z 3 + 9z 2 − 6z − 40 as a product of linear factors. 11. Write each of the following polynomials P (z) as a product of linear factors, and hence determine all solutions of P (z) = 0 (over C). (a) P (z) = z 2 − 4z + 8 (b) P (z) = 16z 4 − 9 (c) P (z) = 4z 4 − 16z 2 − 9 (d) P (z) = z 4 − 2z 2 + 1 12. One solution of the equation 4z 3 − 41z 2 + 174z − 41 = 0. is z = 5 − 4i. Find all the solutions of this equation (over C).