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Math 1508 Name D.Wilson
Section 2.4: Complex Numbers
Please note that:
- An exam will be given on Friday, September 12. There will be no make-ups.
- If you miss three or more workshops, your Test-Out Exams will not be scored.
- (a) If a and b are real numbers, the number a + bi is the standard form of a complex number.
- If b = 0, the number a + bi = a is a real number. So all real numbers are complex numbers.
- If b 6 = 0, the number a + bi is called an imaginary number.
- A number of the form bi, where b 6 = 0, is called a purely imaginary number.
(b) If a is a positive number, then the principal square root of the negative number −a is defined to be
−a =
a i.
(c) The numbers of the form a + bi and a − bi are called complex conjugates.
- Write the complex numbers in the standard form a + bi.
(a) 3 +
(b) i, i^2 , i^3 , i^4 , i^5
(c) − 4 i^2 − 3 i
- Perform the operations and write the answers in standard form.
(a) (8 +
2 i)
(b) (6 − 2 i)(2 − 3 i)
(c) (1 − 2 i)^2 − (1 + 2i)^2
- Write the quotients in standard form.
(a)
2 i
(b)
6 − 7 i 1 − 2 i
Section 2.5: Zeros of Polynomial Functions
- (a) The Fundamental Theorem of Algebra...If f (x) is a polynomial of degree n, where n > 0, then f has at least one zero in the complex number system.
(b) Linear Factorization Theorem...If f (x) is a polynomial of degree n, where n > 0, then f has precisely n linear factors, and
f (x) = a(x − c 1 )(x − c 2 )...(x − cn),
where c 1 , c 2 , ..., cn are complex numbers.
(c) The Rational Zero Test...If the polynomial
f (x) = an xn^ + an− 1 xn− 1 + ...a 2 x^2 + a 1 x + a 0
has integer coefficients, then every rational zero of f has the form
p q
, where p and q have no common factors other than 1, and
p q
factor of the constant term a 0 factor of the leading coefficient an
(d) Complex Zeros Occur in Conjugate Pairs...Let f (x) be a polynomial func- tion that has real coefficients. If a + bi, where b 6 = 0, is a zero of the function, then conjugate a − bi is also a zero of the function.
- Use the given zero to find all the zeros of the function, and write the polynomial as a product of linear factors.
(a) f (x) = x^3 + x^2 + 9x + 9, x = 3i
(b) f (x) = x^3 − 7 x^2 − x + 87, x = 5 + 2i
