Unformatted text preview: s in sign of
a polynomial function. For example, consider f (x) = 2x4 + 4x3 − x2 − 6x − 3. If we focus on only
the signs of the coeﬃcients, we start with a (+), followed by another (+), then switch to (−), and
stay (−) for the remaining two coeﬃcients. Since the signs of the coeﬃcients switched once, we say
f (x) has one variation in sign. When we speak of the variations in sign of a polynomial function,
f , we assume the formula for f (x) is written with descending powers of x, as in Deﬁnition 3.1, and
concern ourselves only with the nonzero coeﬃcients.
2 More appropriately, this equation is ‘quadratic in form.’ Carl likes to call it a ‘quadratic in disguise’ because it
reminds him of The Transformers. 212 Polynomial Functions Theorem 3.10. Descartes’ Rule of Signs: Suppose f (x) is the formula for a polynomial
function written with descending powers of x.
• If P denotes the number of variations of sign in the formula for f (x), then the number of
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