Download Fundamental Theorem of Algebra

Precalculus Lesson #20 Mr. Jeckovich
Fundamental Theorem of Algebra: A polynomial function of degree n has ____ zeros in the complex
number system. (this may include: real, complex or repeated roots)
1. Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing
utility to graph the function as an aid in finding the zeros and as a check of your results.
4
3
2
h x = x + 6 x + 10 x + 6x + 9
2. Find a polynomial function with integer coefficients that has the given zeros.
4, 3i , - 3i
3. Use the given zero to find all the zeros of the function.
Function
3
2
fx = x + x + 9x + 9
Zero
r = 3i
4. Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of
linear and quadratic factors that are irreducible over the reals, and (c) in complete factored form.
4
3
2
2
f x = x - 3 x - x - 12x - 20 (Hint: One factor is: x + 4 ).
Properties of Logarithms
Recall: loga x = y  a
y
=x
1. "One -to-one" Property: loga u = loga v  u = v
8
6
4
2
-8 -7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7 8
x
-2
-4
-6
-8
x
2. "Definition" Property: loga a  = x
3. Product Rule:
Proof of Product Rule
Let: loga u = x 
loga v = y
4. Quotient Rule:
5. Power Rule:
6. Change-of- Base Formula:
loga x =
Proof of Quotient Rule
Proof of Power Rule
Proof of Change-of- Base Formula
Let: y = loga x
Examples: Use the change in base formula to write the given logarithm as a multiple of a common logarithm.
1. log4 10 =
2. ln 5
Use the change in base formula to write the given logarithm as a multiple of a natural logarithm.
3.
log4 10 =
4.
log10 5 =
Evaluate the logarithm using the change-of-base formula. Round your answer to 3 decimal places.
5. log7 4 =
6. log20 125 =
Use the properties of logarithms to write the expression as a sum, difference, and/or constant multiple of
logarithms. ( Assume all variables are positive. )
7. ln
x®
y
3
=
8.
4
logb  x  y 
4
z
Write the expression as the logarithm of a single quantity
9. 3lnx + 2ln y - 4ln z
11.
3  6 3  4
ln  5t  - ln  t 
2   4  
10. .5 lnx + 1 + 2lnx - 1 + 3lnx
12. Evaluate without a calculator:
a. log5 1/125 =


b. log4 2 + log4 32 =
4
c. 3ln e 
Assignment
20.) p.194 # 93
p.202 # 3,7,21,29,55,75,95
p.210 # 1,5,9,11,29,37,41
p.259 # 5,13,23,31,33,43
p.267 # 1,3,7,11,19,21,23,41,43,45