quiz6solutions - MAC1147 Quiz#6 In the top-right corner of...

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Unformatted text preview: MAC1147: Quiz #6 10/06/2009 In the top-right 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 x-intercept 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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