9.2 Polynomial Functions
Definitions
1. A function of the form p x = an x n + a n"1 x n"1 +... + a2 x 2 + a1 x + a0 where each exponent is a
positive integer and each ai is a real number is called a polynomial.
* So a polynomial is a bunch of power functions added together
* The polynomial is in standard form if the exponents decrease left to right
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2. The largest exponent is called the degree of the polynomial.
3. Each power function ai x i is called a term of the polynomial.
4. The constants an , a n"1 , ..., a 2 , a1 , a0 are called coefficients.
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5. Assuming n is the largest exponent, the term an x n is called the leading term.
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Long-Run Behavior
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The long-run behavior of a function is basically
its behavior as we move away from the
origin.
The long run behavior of polynomials is its leading term.
So two polynomials that have the same leading term will look alike in the long run.
Ex1
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f x = x 3 + x 2 and g x = x 3
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Ex2
f x = 0.5x 4 " 2.5x 2 + 2 and g x = 0.5x 4
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f x = x 4 " 4x 3 +16x "16
Ex3
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g( x ) = x
h( x ) = x
4
" 4x 4 " 4x 2 +16x
4
+ x 3 " 8x 2 "12x
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** Long run behavior – two different functions can look alike
** Short run behavior (closer to the origin) – these functions are clearly different