HONORS Algebra Two, Polynomial Unit Day 5
Name:________________________ Date: _____________
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Rational Zero Theorem - All rational zeros (roots, solutions, answers) are in the form
where
q
p is a factor of the constant term and q is a factor of the leading coefficient.
Rational Zero Theorem helps you determine possible rational zeros for a function.
---------------------------------------------------------------------------------------------------------------------------------ex 1 - List all possible rational roots (zeros) for the function f ( x)  2 x3  3x3  6 x  9 .
(a) Which is the constant term? ______ What are all factors of the constant term? _____________
(b) Which is the leading coefficient? _______ List all of its factors. ______________
p 
(c) List all possible rational zeros  ' s  .
q 
Ans:

_____________________________
ex 2 - List all possible rational roots (zeros) for the function f ( x)  x 4  7 x3  9 x 2  7 x  10 .
(a) Which is the constant term? ______ What are all factors of the constant term? _____________
(b) Which is the leading coefficient? _______ List all of its factors. ______________
p 
(c) List all possible rational zeros  ' s  .
q 
Ans:

_____________________________
ex 3 - List all possible rational roots (zeros) for the function f ( x)  x3  3x 2  6 x  8 .
ex 4 – You can uses synthetic substitution to help determine zeros of a function. If you plug in a number that
is a zero of the function, you should get a zero in that final spot of the substitution.
Find all zeros (roots) of f ( x)  x3  3x 2  6 x  8 given that one root is x  4 .
ex 5 – Use the same procedure as ex 4.
Find all zeros (roots) of f ( x)  x3  7 x 2  16 x  10 given that one root is x  1 .
Ex 6 – Find all roots of f ( x)  x3  4 x 2  x  6 . (Hint: For this problem, you must guess one of the roots
p
using ' s . You may not guess correctly on the first attempt, so be patient. Okay, you’re probably not going
q
to be patient, so let the graphing calcuator assist you.)
Step 1: List all
p
's.
q
Step 2: Choose one of the
p
' s and do synthetic division to see if it works. Repeat this process until
q
you find one that works.
Step 3: Write the “depressed polynomial”.
Step 4: If the “depressed polynomial” is second degree, factor to find the answers.
Ex 7 – Find all roots of f ( x)  x 4  3x 2  2 .
Algebra Two
Name: _______________________ Date: _____________
p 
' s  and FINDING ZEROS WITH SYNTHETIC DIVISION
q 
HOMEWORK: 
p 
List all possible rational roots  ' s  and find all roots (zeros) of the problems. Use your GC for help.
q 
1. f ( x)  x3  3x 2  x  3
2.
f ( x)  x 4  7 x3  17 x 2  17 x  6
3.
f ( x)  x3  x 2  14 x  24
4.
f ( x)  x3  3x 2  9 x  13
(Hint: this one has one Real root and 2 imaginary roots.)