Print Name: _________________________
MathB1a / Spring 2011
Exam 3
Chapter 3
Read Carefully: Relative work must be shown in order to receive credit. Simplify all answers
(no calculator approximations unless specified). Partial credit is rewarded, so it is to your benefit
to show steps. If not specified, you may use your graphing calculator to check work, but proper
algebraic steps must be shown to derive answers. Clearly label answers. If space provided is
insufficient, additional paper will be provided. Be sure to clearly label the problems on any
additional sheets used. Good luck!
You will need the following theorems:
Rational Zeros Theorem
If the Polynomial P( x ) = a n x n + a n −1 x n −1 + + a 1 x + a 0 has integer coefficients, then
every rational zero of P is of the form
p
q
where p is a factor of a0 & q is a factor of an.
Descartes’ Rule of Signs
Let P be a polynomial w/ real coefficients.
1. The number of positive real zeros of P(x) is either equal to the # of variations in sign of
P(x) or is less than by an even whole #.
2. The number of negative real zeros of P(x) is either equal to the # of variations in sign of
P(–x) or is less than by an even whole #.
The Upper/Lower Bounds Theorem
Let P be a polynomial with real coefficients
1. If we divide P(x) by x – b (b > 0) using synthetic division, & if the third row has no
negative entry, then b is an upper bound for the real zeros of P.
2. If we divide P(x) by x – a (a < 0) using synthetic division, & if the third row has entries
alternately nonpostive & nonnegative, then a is a lower bound for the real zeros of P.
Exam 3 – version 2
1
1) f ( x ) = ( x + 3)( x − 2) 3 ( x − 5) 2
a) Complete the table:
Leading term
Zeros and their
multiplicity
b) Sketch the graph.
Exam 3 – version 2
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2) The graph below represents a polynomial function.
y
1
−3
x
1
6
Find a polynomial of degree 4 whose graph is shown above.
(You may leave answer in factored form).
Exam 3 – version 2
3
3) Find a polynomial P(x) of degree 3 with integer coefficients that has zeros 1 and 3 –2 i.
(Write answer in standard form)
Exam 3 – version 2
4
4) Let
P( x ) = x 4 − 7 x 3 + 25x 2 + 11x − 150 .
a) Given that f(x) has zeros 3 and –2, find the remaining zeros.
b) Factor P into linear and irreducible quadratic factors with real coefficients.
c) Factor P completely into linear factors with complex coefficients.
Exam 3 – version 2
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5) Use the Rational Zero Theorem to list all the possible rational zeros of the following
polynomial. DO NOT NEED TO FIND ZEROS.
f ( x ) = 9 x 5 − 21x 4 − 50 x 3 + 130 x 2 − 24 x − 24
Exam 3 – version 2
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6) P(x) = x3 + 7x – 5
a) Use the Rational Zero Theorem to list all the possible roots of P(x).
b) Show P has no rational zeros.
c) Show the conditions of The Intermediate Value Theorem are met that guarantee at
least one zero exists on the interval [0, 1].
Exam 3 – version 2
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7) Let P( x ) = x 3 − 8x 2 + 21x − 18
(You will apply the results on this page for problem #8)
Answer the following:
a) Use the Rational Zero Theorem to list all the possible roots of P(x).
b) Use Descartes’ Rule of Signs to fill in the blank:
The number of possible positive real roots: ____________
The number of possible negative real roots: ____________
c) Does The Upper/Lower Bound Theorem verify that 9 is an upper bound of the
zeros for P(x)? (Be sure to show work and answer YES or NO)
Exam 3 – version 2
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8) Given the polynomial from problem 7), , P( x ) = x 3 − 8x 2 + 21x − 18 , do the
following:
a) Using all your answers to problem 7 (must include parts a, b and c), what are the
possible zeros of P(x).
b) Find all the zeros of P(x)
Exam 3 – version 2
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9) Identify the horizontal asymptotes for the following. If no horizontal asymptote exists,
State “NO HORIZONTAL ASYMPTOTE” .
(Do Not Need to Graph)
a) r ( x ) =
b) r ( x ) =
Exam 3 – version 2
5x 4 − x + 2
4 x 3 − 6x + 3
2 x 3 − 3x 2 + 3x + 4
x 4 − 3x + 1
10
2 x 2 − 50
x 2 − 6x − 7
a) Find the x-intercepts.
10) r ( x ) =
b) Find the y-intercepts.
c) What are the vertical asymptotes:
d) What are the horizontal or oblique asymptotes.
e) Graph. Be sure the information from parts a) through d) are clearly labeled on the
graph. (You may plot some more points to help).
Exam 3 – version 2
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