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the x-intercepts of the graph of a function and the zeros of a function, the Factor Theorem, the
role of multiplicity, complex conjugates, the Complex Factorization Theorem, and end behavior of
polynomial functions. (In short, you’ll need most of the major concepts of this chapter.) Since the
graph of p touches the x-axis at 3 , 0 , we know x = 1 is a zero of even multiplicity. Since we
are after a polynomial of lowest degree, we need x = 3 to have multiplicity exactly 2. The Factor
Theorem now tells us x − 1 is a factor of p(x). Since x = 3i is a zero and our ﬁnal answer is to
have integer (real) coeﬃcients, x = −3i is also a zero. The Factor Theorem kicks in again to give us
(x − 3i) and (x +3i) as factors of p(x). We are given no further information about zeros or intercepts
so we conclude, by the Complex Factorization Theorem that p(x) = a x − 3 (x − 3i)(x + 3i) for
some real number a. Expanding this, we get p(x) = ax4 − 23 x3 + 82a x2 − 6ax + a. In order to obtain
integer coeﬃcients, we kno...
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