A polynomial function has a root of negative 7 with multiplicity 2 , a root of negative 1 with multiplicity 1 , a root of 2 with multiplicity 4 , and a root of 4 with multiplicity 1 If the function has a positive leading coefficient and is of even degree, which statement about the graph is true?
Answer
A
The graph of the function is negative on open paren 4 comma infinity close paren .
B
The graph of the function is positive on open paren 2 comma 4 close paren .
C
The graph of the function is negative on open paren negative 7 comma negative 1 close paren .
D
The graph of the function is positive on open paren negative infinity comma negative 7 close paren .
Explanation
The roots at −7 and 2 have even multiplicities, so the graph does not change sign as it passes through them. The roots at −1 and 4 have odd multiplicities, so the graph changes sign at each. Since the leading coefficient is positive, the graph is positive to the right of 4; moving left across 4 makes it negative, and moving left across −1 makes it positive again. Thus the graph is positive on (−∞, −7), so the correct statement is that it is positive there.
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