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Linear Algebra 1
Problem Set 3 (Complex Numbers and Polynomials)
1. Find the algebraic form of the numbers:
π
π
π
π
(a) (cos − i sin )10 ,
(b) (sin + i cos )24
4
4
6
6
2. Use the exponential form of a complex number to solve the following equations
(graph the solutions in the complex plane):
(a) z 3 = z̄ ,
(b) |z|3 = iz 3 ,
(c) |z 8 | = z 4 ,
3. Without carrying out the division, find the remainder of division of P by Q:
(a) P (x) = x30 + x − 1, Q(x) = x2 − 1
(b) P (x) = x40 + x + 1, Q(x) = x2 + 1.
In the following problems, you may find this online utility handy in checking your
solution.
4. Use the Horner algorithm to evaluate the polynomial P at x = a:
(a) P (x) = x3 + 4x2 − 3x + 5, a = 4
(b) P (x) = x6 − 4x5 + 3x4 + 5x3 + 4x2 + 2x − 6, a = −3
5. Use the Horner algorithm to find the quotient and the remainder of division of P by
x − a:
(a) P (x) = x3 + 2x2 + 6x − 2, a = 2
(b) P (x) = x4 + 1, a = −1
6. Guess one of the roots of P . Then divide P by x − a, where a is the guessed root.
Continue until you find all the roots of P . Then write P as a product of linear factors
of the form x − xk (possibly complex).
(a) P (x) = x3 + 3x2 − 4
(b) P (x) = x3 − 2x2 − 3x + 10
(c) P (x) = x4 − 2x3 + 2x2 − 2x + 1
An online utility for polynomial division can be found here.
7. The number x1 is known to be a root of the polynomial P . Find the remaining roots.
Then write P as a product of irreducible real polynomials.
(a) P (x) = x4 − 6x3 + 5x2 + 2x − 10, x1 = 1 + i
(b) P (x) = x4 − 2x3 + 7x2 + 6x − 30, x1 = 1 − 3i
8. The numbers 1 − i and 2 + i are known to be roots of the polynomial
z 6 − 6z 5 + 12z 4 − 35z 2 + 54z − 30,
Find the remaining roots. Then write P as a product of irreducible real polynomials.
9. Write the polynomial P as a product of irreducible real polynomials:
(b) P (x) = x6 − x2 ,
P.Kajetanowicz
(b) P (x) = x6 + 1,
(c) P (x) = x8 − 1