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Unformatted text preview: he Factor Theorem and The Remainder Theorem 3.2 197 The Factor Theorem and The Remainder Theorem Suppose we wish to ﬁnd the zeros of f (x) = x3 + 4x2 − 5x − 14. Setting f (x) = 0 results in the
polynomial equation x3 + 4x2 − 5x − 14 = 0. Despite all of the factoring techniques we learned1
in Intermediate Algebra, this equation foils2 us at every turn. If we graph f using the graphing
calculator, we get The graph suggests that x = 2 is a zero, and we can verify f (2) = 0. The other two zeros seem to
be less friendly, and, even though we could use the ‘Zero’ command to ﬁnd decimal approximations
for these, we seek a method to ﬁnd the remaining zeros exactly. Based on our experience, if x = 2
is a zero, it seems that there should be a factor of (x − 2) lurking around in the factorization of
f (x). In other words, it seems reasonable to expect that x3 + 4x2 − 5x − 14 = (x − 2) q (x), where
q (x) is some other polynomial. How could we ﬁnd such a q (x), if it even...
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