Week 3: Complex Numbers – Day 3

Dividing Complex Numbers by Complex Numbers

Addition and multiplication of complex numbers is straightforward. If you divide by a real number it is easy: divide real and non real parts by the number, for example: (2 + 6i)/2 = 1 + 3i.

Division by a non-real number takes a little more work. The trick is to turn division into multiplication, using the complex conjugate.

Example: (2 + 3i) / (1 – i)

Multiply numerator and denominator by the complex conjugate of 1 – i, which is 1 + i:

(2 + 3i) x (1 + i ) = 2 – 3 + 3i + i = -1 + 4i

(1 – i) x (1 + i) = 1 – i + i + 1 = 2

Solution: (-1 + 4i)/2 = -1/2 + 2i

In the following problems, do the division.

1. 1 / i

(1 divided by i).

2. (2 + 3i)/i

3. (1 + i) / (1 – i)

4. i / (1 – i)

5. (3 – 2i) / (3 + 2i)

Week 3: Complex Numbers – Day 2

Complex Solutions

1. What are the solutions of x2 + 9 = 0?

2. What are the solutions of (x + 1)2 + 4 = 0?

3. What are the solutions of x2 – 2x + 10 = 0?

4. What are the solutions of x3 + 25x = 0?

5. What are the solutions of (x – 1)3 + 36(x – 1) = 0?

Week 3: Complex Numbers – Day 1

Review of arithmetic of complex numbers

Simplify these expressions. The final result should be a number of the form A + Bi.

1. (-1 + i) + (3 + 2i)

2. (7 + 2i) + 10

3. (2 + 2i) x 3i

4. (5 + 3i) x (5 – 3i)

5. (4 + i) x (1 + 6i)

Week 2: Equations – Day 5

Deriving Equations

1. Set up the equation for solving for the intersection between the line joining (0, 0) and (1, 4) and the line x + y = 1.

2. Set up the equation for solving for the zeros of the function f(x) = 2x3 + 4x2 – 17x – 3.

3. Set up the equation for solving for the time it takes to drive 432 kilometers at a speed of 65 kilometers per hour.

4. An isosceles triangle has two angles that are the same: 15 degrees. Set up the equation for solving for the other angle of the triangle.

5. Set up the equation for solving for the principal amount of a loan, if the amount of interest owed in one year is \$700 and the interest rate is 5% per year.

Week 2: Equations – Day 4

Predicting Solutions

Without solving these equations, predict the maximum and minimum number of solutions:

1. 3x + 54 = 8x – 6

2. 4x – 7 = 4x + 20

3. 5 + 4x = 3x2 + 6x

4. 10x5 – 47x4 + 3x3 – x2 + 8 = 0

5. 1 = x2 + 3

Week 2: Equations – Day 3

Equations Involving Exponential Expressions

Exponential expressions have two basic parts: exponent and base. The simplest equations involve one or the other.

1. If 2300 x 2n = 2501 what is n?

2. If 2100 x 3100 = a100 what is a?

3. If 100n x 53 = 20n x 563 what is n?

4. If 249 x 2n = 1/4, what is n?

5. If (2n)10 = 326 what is n?

Week 2: Equations – Day 2

Review of easy equation-solving techniques.

1. How many solutions are there to:

1/(x2 + 1) = 0

2. What are the solutions of:

(x – 2)/(x2 + 1) = 0

3. What are the solutions of:

x2 – 28 = 3x

4. What are the solutions of:

(x + 3)2 = 25

5. Solve:

Week 2: Equations – Day 1

Solve for x:

1.   3x – 3 = 5 – x

2.   1/2 – 1/x = 1/x

3. x4 – 1 = 0

4. 5x2 – 3 = 2x

5. (x2 – 2x – 3)/(x + 1) = 0

Week 1: Polynomials – Day 5

1. What is the domain of a polynomial function?

2. What is the range of a quadratic (degree 2) polynomial function?

3. What is the range of a cubic (degree 3) polynomial function?

4. True or false: the product of two polynomials is a polynomial.

5. True or false: the sum of two polynomials is a polynomial.

Week 1: Polynomials – Day 4

Cubic polynomials.

1. Sketch the graph of f(x) = x3.

2. Sketch the graph of g(x) = x(x2 + 1).

Bonus questions: what is the difference between f(x) and g(x), in the neighborhood of x = 0? How many real roots do f(x) and g(x) have?

3. Sketch the graph of h(x) = x(x – 3)2. Hint: how many real roots does h(x) have? What are the values of h(x) around and between the roots?

4. Sketch the graph of p(x) = x(x – 1)(x – 2). Hint: how many real roots does p(x) have? What are the values of p(x) around and between the roots?

5. Sketch the graph of q(x) = -x(x – 1)(x – 2).