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**Unformatted text preview: **2 0 3.2 The Factor Theorem and The Remainder Theorem 203 1
From the ﬁrst division, we get 4x4 − 4x3 − 11x2 + 12x − 3 = x − 2 4x3 − 2x2 − 12x + 6 . The
3 − 2x2 − 12x + 6 = x − 1
2 − 12 . Combining these results, we
4x
second division tells us 4x
2
4 − 4x3 − 11x2 + 12x − 3 = x − 1 2 4x2 − 12 . To ﬁnd the remaining zeros of p, we set
have 4x
2
√
4x2 − 12 = 0 and get x = ± 3. A couple of things about the last example are worth mentioning. First, the extension of the
synthetic division tableau for repeated divisions will be a common site in the sections to come.
Typically, we will start with a higher order polynomial and peel oﬀ one zero at a time until we are
left with a quadratic, whose roots can always be found using the Quadratic Formula. Secondly, we
√
√
√
found x = ± 3 are zeros of p. The Factor Theorem guarantees x − 3 and x − − 3 are
both factors of p. We can certainly put the Factor Theorem to the test and continue the synthetic
division tableau from above t...

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