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Unformatted text preview: MAC1147: Quiz #6
10/06/2009
In the topright corner of a clean sheet of paper, write your name, UFID,
and section number. Please use a pen with blue or black ink. When you are
nished, FOLD your paper in half lengthwise and write your name on the
back.
1. Let f (x) = x3 − 4x2 + 5x − 2.
(a) Use the Leading Coecient Test to determine the behavior of f (x)
as x → −∞ (rise or fall?) and x → ∞ (rise or fall?).
Since the degree of f (x) is odd, it will rise in one direction and fall
in the other. The fact that the leading coecient is positive tells
us that f (x) rises as x → ∞ and f (x) falls as x → −∞.
(b) Factor f (x) into linear factors (Hint: Use synthetic division with
x = 2).
The synthetic division works out as:
2
1 −4 5 −2 2 1 −4 2 −2 1 0 Hence, x3 − 4x2 + 5x − 2 = (x − 2)(x2 − 2x + 1) = (x − 2)(x − 1)2 .
(c) Use part (b) to state each zero of f (x) and give its multiplicity.
x − 2 = 0 ⇒ x = 2 with multiplicity 1.
(x − 1)2 = 0 ⇒ x = 1 with multiplicity 2.
(d) Use parts (a) and (c) to sketch a graph of f (x). Clearly label (or
list o to the side) each xintercept and the y intercept.
1 2. Let z = 1
.
2−i (a) State the standard form of a complex number. z = a + bi
(b) Use a complex conjugate to rewrite z in standard form.
2 − i = 2 + i. So z = 2+i
2+i
1
·
=
=
2−i 2+i
4 − (−1) (c) TRUE or FALSE: 5 is a complex number. 2 2
5 + 1i
5 ...
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 Summer '08
 GERMAN
 Calculus, Fundamental Theorem Of Algebra, Synthetic Division, Complex number, Root of a function

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