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Unit 3 Lesson 7 Do Now
You have ten minutes to complete the following 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:
Unit 3 Lesson 7 Do Now
You have ten minutes to complete the following 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 Honors
Unit 3 Lesson 7: Synthetic Division with Complex Zeros
Objective: _____________________________________________________________
1. Guided Practice: 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. Skills you need to remember for Algebra 2: Operations on Complex
Numbers
a.)i = _____ b.)i 2 = _____ c.)(1+ i) + (-2 + 3i) = __________ d.)(1+ i)(-2 + 3i) = __________
3. Guided Practice: Synthetic Division with Complex Numbers (Do work on a
separate sheet of paper)
Example 1 (Part 1): The complex number z = 1 – 2i is a zero of
f ( x)  4 x 4  17 x 2  14 x  65 . Do synthetic division to begin to factor this polynomial.
Example 1 (Part 2): Do synthetic division with the complex conjugate to factor this
polynomial completely.
Example 1 (Part 3): Once you do synthetic division with your complex conjugate, use
write f(x) in complete factored form.
Factored Form: ________________________________________________
Example 1 (Part 4): Find all zeros of the polynomial. State whether the zeros are
rational, irrational, or complex.
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Unit 3 Lesson 7 Problem Set
Synthetic Division with Complex Zeros
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. Perry claims that 3 is not a zero of the polynomial below. Janice claims 3 could be a
zero of the polynomial, depending on the value of a. Who is correct and why? Support
your argument with mathematical terminology learned in class.
2x4 + ax3 + 3x2 - 5x + 10
3. Find k such that f (x) = x 4 - kx 3 + kx 2 - 2 has a factor of (x + 2). Explain how you
arrived at your answer.
4. 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.
5. 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.
6. 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.
Pre-Calculus (H) Homework: Finish Unit 3 Lesson 7 problem set
and page 234 #(27, 29, 33, 34)