Stitz-Zeager_College_Algebra_e-book

# The graph of y hx the graph of y rx our next example

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Unformatted text preview: etween 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 1 graph of p touches the x-axis at 3 , 0 , we know x = 1 is a zero of even multiplicity. Since we 3 1 are after a polynomial of lowest degree, we need x = 3 to have multiplicity exactly 2. The Factor 2 Theorem now tells us x − 1 is a factor of p(x). Since x = 3i is a zero and our ﬁnal answer is to 3 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 12 so we conclude, by the Complex Factorization Theorem that p(x) = a x − 3 (x − 3i)(x + 3i) for a some real number a. Expanding this, we get p(x) = ax4 − 23 x3 + 82a x2 − 6ax + a. In order to obtain 9 integer coeﬃcients, we kno...
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