Download U3 L2 I1+ Homework - Mayfield City Schools

Math 4 Honors
U3 L2 I1+: A Complex Solution
Name__________________________
Date ______________________
Learning Goals:
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I can understand the need for complex numbers to solve quadratic equations and the definition of the new
numbers in the form a + bi with a and b real numbers and i  1 .
I can use definitions of addition, subtractions, multiplication, and division of complex numbers to establish
algebraic properties of complex number operations.
I can use synthetic division and the depressed quadratic to find all complex roots of a polynomial function.
I can use a polynomial’s roots to find the equation of the polynomial in standard form with integer
coefficients.
I can find algebraically and classify a polynomial’s zeros.
I can use chunking (“u”-substitution) and factoring to solve complicated equations.
I. At this point in Unit 3, you have come to realize that the degree of a polynomial function tells us how
many zeros the function has. Answer the following questions for functions g & f.
f ( x)  x 3  2 x  4
g ( x)  x 4  3x3  12 x 2  24 x  32
1. How many zeros does each function have?
2. Use your calculator to find all of the exact zeros.
3. What happened in #2?
Notes:
Irrational roots occur in ___________________________________.
i2 = _______
Complex numbers are made up of
Standard form of a complex number: ___________
a is the ________________ part
bi is the ________________ part
Complex roots also occur in ___________________________________.
OVER 
3
We already know the real root of f ( x)  x  2 x  4 . Use the real root & synthetic division to find
the two complex roots.
II. For the following examples, use your calculator, synthetic division & the quadratic formula to find all
exact possible roots of each function.
1. p( x)  x3  7 x 2  12 x  10
2. h( x)  x 4  3x3  x 2  7 x  30
III. Graph the following function in your calculator: f ( x)  x 4  2 x3  6 x 2  2 x  5 .
Any x-intercepts?
What does this tell you about the roots?
We can use imaginary numbers when doing synthetic division.
Verify that i is a root of the function.
If i is a root of the function, what is another root? _______
Use synthetic division to find the other two roots of the function.
IV. Given the following zeros, write polynomial equations in standard form with integer coefficients.
1
1. x  2, 3,
2. x  2  6
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3. x  3,
4
,  7i
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V. Math 2 Honors Review. Calculate the following. Write your exact answers in standard form.
1. (4.5 + 7.2i) + (9 + 3.1i)
2. (12 + 9i) – (-3 – 9i)
3. (5 – 7i)(9 + 3i)
4. (4 + 7i)(4 – 7i)
5.
(–2 + 9i) ÷ (–3 + 2i)
OVER 
U3 L2 I1+ Homework: Please show all work on another piece of paper. You can check
your answers by using the Wolframalpha website!
3. Calculate p + q, p – q, pq, p ÷ q, and qq for the following pair of complex numbers.
p = 1 + 2i and q = 5i – 3
4. When you use the quadratic formula to solve an equation in the form ax2 + bx + c = 0 for x
(assuming a, b, and c are integers), how can you be sure that:
a. the solutions will be rational numbers?
b. the solutions will be irrational real numbers?
c. the solutions will not be real numbers?
5. Powers of the imaginary number i have an interesting property.
a. Using the fact that i2 = -1, calculate i3, i4, i5, i6, i7, i8. Do you notice a pattern?
b. Use your discovery in part a to find a simpler expression for i15and then for i74.
6. DO THIS PROBLEM ON A SEPARATE SHEET OF PAPER. THIS IS TO BE THE ONLY
ITEM ON SAID PAPER – OTHER THAN YOUR NAME.
The complex number system was constructed in stages that began with the set W of whole numbers {1, 2, 3,
4, …} and gradually introduced other important sets of numbers as practical problems and mathematical
problems required them. Create your own graphic organizer (different than the one in this packet) to show the
relationships among the natural numbers, whole numbers, integers, rational numbers, irrational numbers, real
numbers, imaginary numbers, and complex numbers.
7. In solving a new mathematical problem, it often helps to reduce it to a simpler problem by
temporarily ignoring some details. Consider the task of solving the quartic equation
x4 + 6x2 – 16 = 0.
a. Suppose that u = x2. Rewrite the given equation in equivalent form with the letter u.
b. Solve the resulting equation for u.
c. Now use the relationship u = x2 to solve for x. Find ALL solutions.
d. Adapt the substitution strategy suggested in parts a–c to solve these higher-degree equations
and check the solutions you come up with.
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i. 4x4 – 81 = 0
ii. 2x4 + 3x2 – 2 = 0
iii. (log 5 ( x  1))  log 5 ( x  1)  2
***FYI problem #7 involves a HUGE technique used in calculus called u-substitution or chunking!