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Unformatted text preview: g calculator. Example 3.3.6. Let f (x) = 2x4 + 4x3 − x2 − 6x − 3. 1. Find all real zeros of f and their multiplicities. 2. Sketch the graph of y = f (x). Solution. 1. We know from Cauchy’s Bound that all of the real zeros lie in the interval [−4, 4] and that our possible rational zeros are ± 1 , ± 1, , ± 3 , and ± 3. Descartes’ Rule of Signs guarantees 2 2 us at least one negative real zero and exactly one positive real zero, counting multiplicity. We 1 try our positive rational zeros, starting with the smallest, 2 . Since the remainder isn’t zero, 1 we know 2 isn’t a zero. Sadly, the final line in the division tableau has both positive and negative numbers, so 1 is not an upper bound. The only information we get from this division 2 1 1 is courtesy of the Remainder Theorem which tells us f 2 = − 45 so the point 2 , − 45 is 8 8 on the graph of f . We continue to our next possible zero, 1. As before, the only information we can glean from this is that (1, −4) is on the graph of f . When we try our next possible 3 z...
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