Download U3L8 Synthetic Division with Complex Numbers

Pre-Calculus Unit 3 Lesson 8
Name: _____________________________________________
Period # _____
Unit 3 Lesson 8 Do Now
f ( x)  2 x 4  7 x 3  11x 2  28x  12
1.
Find all zeros of the polynomial. State whether each zero is rational, irrational, or
complex.
2. Graph f(x). Make sure to include the zeros, y-intercept, and end behavior in your sketch.
Name: _____________________________________________
Period # _____
Unit 3 Lesson 8 Do Now
f ( x)  2 x 4  7 x 3  11x 2  28x  12
1. Find all zeros of the polynomial. State whether each zero is rational, irrational, or
complex.
2. Graph f(x). Make sure to include the zeros, y-intercept, and end behavior in your sketch.
Pre-Calculus Unit 3 Lesson 8
Pre-Calculus Honors
Unit 3 Lesson 8: Synthetic Division with Complex Zeros
Objective: _____________________________________________________________
1. Do Now: Read and markup the following definition. Use the definition to list the
complex conjugates of the following complex zeros.
The Fundamental Theorem of Algebra States: A polynomial function of a degree n has n
zeros(real and non real). Some of these zeros may be repeated. Every polynomial of odd
degree has at least one zero.
Suppose that f(x) is a polynomial function with real coefficients. If a and b are real
numbers with b  0 and a + bi is a zero of f(x), then its complex conjugate a – bi is also
a zero of f(x).
A.)
-3i
Complex conjugate: _________________________
B.)
1+i
Complex conjugate: _________________________
C.)
3 – 2i Complex conjugate: _________________________
2. Review: Operations on Complex Numbers
a.)i  _________________
b.)i 2  _________________
c.)(1  i )  (2  3i )  __________ d .)(1  i )  (2  3i )  __________
Pre-Calculus Unit 3 Lesson 8
3. Guided Practice: Synthetic Division with Complex Numbers
Directions: The complex number z = 1 – 2i is a zero of
f ( x)  4 x 4  17 x 2  14 x  65 . Find all the zeros of this polynomial and
completely factor the polynomial.
(Part 1): Do synthetic division with the given zero.
SCRATCH
WORK
(Part 2): Do synthetic division with the complex conjugate to break this
polynomial down to a quadratic.
(Part 3): Find all zeros of the polynomial. State whether the zeros are
rational, irrational, or complex.
(Part 4): Write f(x) in complete factored form.
Factored Form: _______________________________________________
Pre-Calculus Unit 3 Lesson 8
Unit 3 Lesson 8 Problem Set
1. The complex number z = 1 + 3i is a zero of f ( x)  x 4  2 x 3  5x 2  10 x  50 . Completely
factor this polynomial and find all the zeros of the polynomial.
2. The complex number z = 3 – 2i is a zero of f (x) = x 4 - 6x 3 +11x 2 +12x - 26 . Completely
factor this polynomial and find all the zeros of the polynomial.
3. The complex number z = 4i is a zero of f (x) = x 4 +13x 2 - 48. Completely factor this
polynomial and find all the zeros of the polynomial.
4. Write a polynomial, with real coefficients that meets the following conditions:



Degree of 4
Roots at x = 3, x = -1, x = 2i
f(0) = 30
5. Is it possible to find a polynomial with a degree of 3 with real number coefficients that has -2
as its only real zero? Explain using terminology learned in class.
6. Is it possible to find a polynomial function of a degree of 4 with real coefficients that has zeros
1+3i and 1-i. Explain using terminology learned in class.
7. Is it possible to find a polynomial function of a degree of 4 with real coefficients that has zeros
-3, 1 + 2i, and 1 - i. Explain using terminology learned in class.