Precalculus Survey Quiz – Section 3.2
NAME_____________________________
1.) The given graph is the graph of the polynomial P(x) = 18x5 + 67x4 – 166x3 – 589x2 – 14x + 24. It has 2
p
integer roots, 1 “other” rational root ( ), and 2 irrational roots.
q
a.) Use the two integer roots (taken from the graph) to perform synthetic division on the given
polynomial to “reduce” the equation to look like: (x + c)(x – k)(ax3 + bx2 + fx + d).
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0
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“Factored” form after synthetic division:
( ________ )( ________ )( ________________)
b.) Use the remaining “cubic” (third degree polynomial) to find the “other” rational root (
p
). List all
q
the factors of the “left over” cubic’s constant term (list the p’s).
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c.) List all the factors of the “left over” cubic’s Leading coefficient (list the q’s).
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d.) List all the possibilities for non-integer rational roots (list all
p
’s that don’t reduce to an integer).
q
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e.) Use synthetic division to find the one remaining rational root of P(x). (You may need scrap paper to
find the CORRECT answer for this part!) List ONLY the correct root and synthetic division.
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0
“Factored” form after synthetic division:
( ________ )( ________ ) ( ________ ) ( ________________)
f.) Use the remaining trinomial and the quadratic formula to find the two irrational roots. Leave your
m± n
“roots” in fractional form, please! (for example,
form).
p
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g.) The 5 roots/zeroes of #1 are
x = _____ _____ _____ and _________
2.) The given graph is the graph of the polynomial Q(x) = 11x3 + 21x2 – 13x + 1. It has no integer roots, 1
p
rational root ( ), and 2 irrational roots.
q
a.)
List all the factors of the cubic’s constant term (list the p’s).
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b.) List all the factors of the cubic’s leading coefficient (list the q’s).
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c.) List all the possibilities for rational roots (list all
p
’s).
q
__________________________________________
d.) Use synthetic division to find the one rational root of Q(x). (You may need scrap paper to
find the CORRECT answer for this part!) List ONLY the correct root and synthetic division.
Then “reduce” the equation to look like: (x + c)(ax2 + bx + d).
_______
_______
_______
_______
_______
_______
________
0
“Factored” form after synthetic division:
( ________ ) ( ________________)
e.) Use the remaining trinomial and “completing the square” or the quadratic formula to find the two
m± n
irrational roots. Leave your “roots” in fractional form, please! (for example,
form).
p
__________________________________
f.) The 3 roots/zeroes of #2 are
x = _____ and _________
Perform the indicated LONG DIVISION. (Each problem worth 9 points)
3.)
4.)
x–3
x4
+ 7 x3
− 7x2
− 43x + 42
x3 – 2x2 x 5
− 3x 4
+ 0x3
+ 4x 2