Unformatted text preview: utting or downright frightening. The
best thing to do is look at an example. Consider f (x) = 4x5 − 3x2 + 2x − 5. Is this a polynomial
function? We can re-write the formula for f as f (x) = 4x5 + 0x4 + 0x3 + (−3)x2 + 2x + (−5).
Comparing this with Deﬁnition 3.1, we identify n = 5, a5 = 4, a4 = 0, a3 = 0, a2 = −3, a1 = 2,
and a0 = −5. In other words, a5 is the coeﬃcient of x5 , a4 is the coeﬃcient of x4 , and so forth; the
subscript on the a’s merely indicates to which power of x the coeﬃcient belongs. The business of
restricting n to be a natural number lets us focus on well-behaved algebraic animals.1
Example 3.1.1. Determine if the following functions are polynomials. Explain your reasoning.
1. g (x) = 1 4 + x3
x 2. p(x) = 4x + x3
x 3. q (x) = 4x + x3
x2 + 4 Enjoy this while it lasts. Before we’re through with the book, you’ll have been exposed to the most terrible of
algebraic beasts. We will tame them all, in time. 180 Polynomial Functions 4. f (x) = √
3 x 5. h(x) = |x| 6. z (x) = 0 Solution.
1. We note directly that the domain of g (x) =...
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