Pre-Calc Worksheet: Real Zeros of Polynomials
Do these problems by hand.
Name__________________________________ Period_________________
1. List all possible rational zeros given by the Rational Root Test (but don’t check to see which
actually are zeros).
a. f ( x)  2 x5  3x3  4 x 2  8
b. p( x)  x 4  3x3  6 x  20
2. Use Descartes’ Rule of Signs to determine how many positive and how many negative real
zeros the polynomial can have.
a. f ( x)  x3  x 2  x  3
positive: _____________________
negative: ____________________
b. f ( x)  2 x6  5x 4  x3  5x  1
positive: _____________________
negative: ____________________
c. f ( x)  x5  4 x3  x 2  6 x
positive: _____________________
negative: ____________________
3. Find integers that are upper and lower bounds for the real zeros of the polynomial. Also use
the Intermediate Value Theorem to identify consecutive integers between which any real zeros
reside. Also use Descartes’ Rule of Signs.
a. p( x)  x3  3x 2  4
UB: ___________
LB: ___________
# poss. (+) zeros: ________
# poss. (-) zeros: _________
zeros will be between: _________
b. . p( x)  x 4  2 x3  x 2  9 x  2
UB: ___________
LB: ___________
# poss. (+) zeros: ________
# poss. (-) zeros: _________
zeros will be between: _________
c. . p( x)  x5  x 4  1
UB: ___________
LB: ___________
# poss. (+) zeros: ________
# poss. (-) zeros: _________
zeros will be between: _________
For 4-12, a) state the number of possible positive and negative real roots. B. List the possible
rational roots. C. Then find all real zeros of the polynomial.
4. f ( x)  x3  3x  2
# poss. (+) zeros: ________
# poss. (-) zeros: _________
p
: ______________________
q
Real zeros: ________________
5. f ( x)  x3  6 x 2  12 x  8
# poss. (+) zeros: ________
# poss. (-) zeros: _________
p
: ______________________
q
Real zeros: ________________
6. f ( x)  x3  4 x 2  x  6
# poss. (+) zeros: ________
# poss. (-) zeros: _________
p
: ______________________
q
Real zeros: ________________
7. f ( x)  x3  4 x 2  3x  2
# poss. (+) zeros: ________
# poss. (-) zeros: _________
p
: ______________________
q
Real zeros: ________________
8. f ( x)  x 4  6 x3  4 x 2  15x  4
# poss. (+) zeros: ________
# poss. (-) zeros: _________
p
: ______________________
q
Real zeros: ________________
9. f ( x)  x3  3x 2  4 x  12
# poss. (+) zeros: ________
# poss. (-) zeros: _________
p
: ______________________
q
Real zeros: ________________
10. f ( x)  2 x 4  3x3  4 x 2  3x  2
# poss. (+) zeros: ________
# poss. (-) zeros: _________
p
: ______________________
q
Real zeros: ________________
11. f ( x)  x 4  5 x3  6 x 2  4 x  8
# poss. (+) zeros: ________
# poss. (-) zeros: _________
p
: ______________________
q
Real zeros: ________________
12. f ( x)  x5  x 4  3x3  5x 2  2 x
# poss. (+) zeros: ________
# poss. (-) zeros: _________
p
: ______________________
q
Real zeros: ________________