PreCalculus: Lesson Plan * 10/9/2006

Quickstart
1. Use synthetic division to determine if (x + #) is a factor of
f(x) = 3x³ + 4x² -18x - 3.
2. Divide using long division
(4x⁵ - x³ + 2x² - x) / ( 2x + 1 )
3. Use the Remainder Theorem to evaluate f(-3) for f(x) = 2x³ - 4x² + 1.

Lesson
2.3 Real Zeros of Polynomial Functions [continued]
Objective: Determine upper and lower bounds for zeros of polynomial functions.

V. Upper and Lower Bound Test
A way of dealing with a very large list generated by the Rational Zero Test is the Upper and Lower Bound Rule.
A real number b is an upper bound for the real zeros of f if there no zeros of f greater than b.
A real number b is a lower bound for the real zeros of f if there no zeros of f less than b.

Upper and Lower Bound Rule
Let f be a polynomial function with real coefficients and a positive leading coefficient. Suppose f(x) is divided by x – c using synthetic division.
1. If c > 0 and each number in the last row is either positive or zero,
then c is an upper bound for the real zeros of f.
2. If c < style="font-style: italic;">


2.4 Complex Numbers
Objective: Students will know how to perform operations with complex numbers and plot complex numbers in the complex plane.

I. Imaginary unit i
A. Definition
We need more numbers because simple equations such as x² + 1 = 0 do not have a real solution. We need a number whose square is –1.
i² = -1. i is called the imaginary unit.
By adding real numbers to multiples of the imaginary unit we get the set of complex numbers, defined as {a + bi | a is real, b is real and i² = -1}.
a + bi is the standard form for a complex number.
a is called the real part and bi is the imaginary part.
B. Two complex numbers a + bi and c + di are equal to each other if and only if a = c and b = d.

II. Operations with Complex Numbers
A. To add two complex numbers, we add the two real parts then add the two
imaginary parts. That is,
(a + bi) + (c + di) = (a + c) + (b + d )i


III. Complex Conjugates and Division
Two complex numbers in the form a + bi and a - bi are called complex conjugates.
(a + bi)(a - bi) = a² - (bi)² = a² + b².
We can use this fact to divide complex numbers.


Homework
p. 180 - 181 # 10 - 60 x 5's