Review: complex numbers

Transcription

1 October 5/6, 01.5 extra problems page 1 Review: complex numbers Number system The complex number system consists of a + bi where a and b are real numbers, with various arithmetic operations. The real numbers are a subset of the complex numbers: a = a + 0i. Operations To add or subtract complex numbers, just combine like terms. To multiply complex numbers, use the distributive property and the fact i = 1. To divide complex numbers, use this technique involving a conjugate: a bi c di c di c di A square root of a + bi is a complex number whose square equals a + bi. Every non-zero complex number has two square roots, which are opposites of each other. For example, ( 3i) = 5 1i and ( + 3i) = 5 1i. As of now, we don t have a definition for how to use a complex number as an exponent. That requires some trigonometry ideas that we haven t established yet. Other operations on a complex number z = a + bi: absolute value (or modulus) z = a b conjugate z = a bi (we ll work with this later in the year.)

2 October 5/6, 01.5 extra problems page Part I: complex numbers 1. Show that the sum of a + bi and its conjugate is a real number.. Show that the product a + bi and its conjugate is a nonnegative real number. 3a. How could you show that 79 is the square root of 641 without using a calculator? 3b. How could you show that 3 i is a square root of 8-6i? 4. Show that 4-3i is a square root of 7-4i. 5. Show that 1 i is a square root of i. 6. Find the square roots of 3+4i.

3 October 5/6, 01.5 extra problems page 3 Write the following in the form a + bi i i 9. 3 i 7 i 10. i 5 3 i i i i i i i i Part II More Practice with GCF and LCM 13. f(x) = x 4 8x 3 + 7x 50x + 50 and g(x) = 3x 5 6x x 3 44x 86x Find GCF(f(x), g(x)) and LCM(f(x), g(x)) given the hint that f(1 i) = 0. Definition: A pair of numbers or a pair of polynomials is called relatively prime if the pair has a greatest common factor (GCF) of Suppose f x and gx are polynomials. The zeros of f x are 3, 1, and 5. The zeros of gx are 4, 1, and 1. Are f x and gx relatively prime? Explain. 3 m and n x x 5x 8x 6. Find all the zeros (real and non-real) for both functions given the hint that m x and n x are not relatively prime Suppose x x x 4x 4x 4

4 October 5/6, 01.5 extra problems page 4 Part III: Problems about solutions and factors of polynomial functions 1. Consider the equation x a. How many solutions does this equation have in the real number system? b. According to the Fundamental Theorem of Algebra, how many solutions does this equation have in the complex number system? c. Find all solutions of the equation in the complex number system. a. Find a quadratic polynomial with real coefficients having ( + 5i) as a zero. Write your answer in standard form (that is, multiply the factors, don t leave in factored form). b. Find a cubic polynomial with real coefficients having 4 and (3 i) as zeros. Write your answer in standard form. Hint: Easiest if you multiply the factors from the non-real zeros first. 3. To answer these questions you will need to write polynomials that have complex numbers as coefficients. It is not possible to write answers having real coefficients. a. Find a linear polynomial with complex coefficients having ( + 5i) as a zero. b. Find a quadratic polynomial with complex coefficients having 4 and (3 i) as zeros. Write your answer in standard form. 4. You are given this information about polynomial P(x): P(x) is a degree 5 polynomial with real coefficients. The graph of P(x) for real numbers x is given on the grid. Its intercepts are at x = 3, x = 4, and y =. In the complex numbers, P(i+1) = 0. a. When P(x) is divided by (x 3), what is the remainder? b. Factor P(x) completely in the complex number system. c. Factor P(x) as completely as possible in the real number system. 5. Suppose f(x) = x 4 8x 3 + 7x 50x a. Verify that f(1 i) = 0. b. Factor f(x) completely in the complex number system. c. Factor f(x) as completely as possible in the real number system.

5 October 5/6, 01.5 extra problems page 5 Problems about incorrect uses of theorems Directions: The following argument reaches an incorrect conclusion. Identify the erroneous step in the argument, identifying which theorem was misused and explaining why the use is incorrect. Then, answer the follow-up question. 6. Argument: Let P be a polynomial with complex coefficients, having + 3i as a zero. Since non-real zeros occur in conjugate pairs, P cannot be a polynomial of degree 1. Follow-up question: Find polynomials of degrees 1,, and 3, each having + 3i as a zero. Make them polynomials with real coefficients, if possible. Problems about square roots and complex zeros 7. a. Find the square roots of ( i). That is, find a + bi having the property (a + bi) = ( i). b. Check your answer by typing ( i) on your calculator. 8. Consider this polynomial with complex coefficients: f(x) = x + 3i x + (1 i). a. Use the Quadratic Formula to find the zeros of f(x). Hint: The result of problem 7 will be needed. b. Write a complete factorization of f(x). ANSWERS: Parts I and II: 1. a. a b 0 3. a i 7 4i 79 b. 3 i 8 6i 5. 1 i i i and i 7. i 8. i i 10. i 11. i 1. 1 i GCF: x - x + 5; LCM: (x - x + 5)(x - 6x + 10)(3x 3-8x -5x + 6) 14. No - because (x - 1) is a common factor for both functions. 15. m(x): x 1 i, x i n(x): x = 3, x 1 i

6 October 5/6, 01.5 extra problems page 6 Part III Depending on the problem, what s shown here might be a complete solution, or just the final answer, or just a hint or a first step. We ll go over questions on Monday. I can also answer give some help by when I am away on Thursday and Friday. 1. c. 3, i, 3 i. a. x + 4x + 9 b. Begin from (x 4)(x (3 i)) (x (3 + i)). The easiest start is to multiply the second and third factors. That gives (x 4)(x 6x + 13). Then use the distributive property to get x 3 10x + 37x a. x ( + 5i) b. (x 4)(x (3 i)), then either distribute or apply the formula from a, to get x + ( 7 + i)x + (1 8i). 4. a. remainder = P(3) = 5 b. 1 (x + 3) (x 4)(x (1 + i))(x (1 i)) c (x + 3) (x 4)(x x + ) 5. b. (x (1 i))(x (1 + i))(x (3 + i))(x (3 i)) c. (x 6x + 10)(x x + 5) 6. Complex Conjugates Theorem does not apply when the coefficients are complex. (x ( + 3i)); x 4x + 13; x 3 4x + 13x 7. a. You need to solve the system a b = 13, ab = 8. Solve the second equation for a, substitute into the first equation. Multiply by b to get b 4 13b 16 = 0. Let x = b and apply the quadratic formula to x 13x 16 to get x = But x = b cannot be negative, so discard the negative possibility Square root again to get b =. Perhaps the easiest way to get a is to repeat the process solving for the other variable. Get a = So a + bi = i or i. 8. a. Quadratic formula gives the square root. 3i 13 8i, then substitute the answers to 7a in place of