Section 5.1 Dr. Doug Ensley Fall 2013
Polynomial Functions A polynomial is a sum of monomials. A monomial is an algebraic expression of the form c · x n , where c is any real number (the coefficient) and n is a non-negative integer. For example, the following functions on the left are polynomials and the functions on the right are not: 1. f ( x ) = − 4 x − 1 1. f ( t ) = t 3 − 1 t 2 3 2 x 2 + 2 x 1 / 2 2. g ( z ) = π z 10 2. g ( x ) = 1 3. h ( x ) = √ x . 3. h ( x ) = 5 x 2 +1 6 2 t 2 + t + √ 4. p ( z ) = z 2 + z 4. p ( t ) = 1 5 z
Factored Polynomial Functions We can tell a lot about a polynomial if we have it in factored form. For example, the following function is a polynomial: f ( x ) = 6( x − 2) 2 ( x 2 + 5) ◮ What is the degree of this polynomial? ◮ What is the domain of this function? ◮ Find the x - and y -intercepts of f .
Factors. zeros, and multiplicity When a polynomial is completely factored, we refer to its zeros and their multiplicities . For example, consider the following function: p ( x ) = 1 2( x − 1) 3 (2 x + 3) 2 (1 2 x + 2) The factors are . . . ◮ x = 1 is a zero with multiplicity 3 ◮ x = − 3 / 2 is a zero with multiplicity 2 ◮ x = − 4 is a zero with multiplicity 1 From this we can tell that the x -intercepts will occur at the points (1 , 0), ( − 3 / 2 , 0), and ( − 4 , 0), but the multiplicities tell us even more.
Multiplicities p ( x ) = 1 2 ( x − 1) 3 (2 x + 3) 2 ( 1 2 x + 2)
Multiplicities p ( x ) = 1 2 ( x − 1) 3 (2 x + 3) 2 ( 1 2 x + 2) ◮ The odd multiplicity of the ( x − 1) factor tells us the graph crosses the x -axis at x = 1. ◮ The even multiplicity of the (2 x + 3) factor tells us the graph touches the x -axis at x = − 3 / 2. ◮ The odd multiplicity of the ( 1 2 x + 2) factor tells us the graph crosses the x -axis at x = − 4.
Custom polynomials Create a polynomial (in factored form) with lead coefficient 1 having these characteristics: ◮ Zeros at − 1, 2, and 3 having degree 3 ◮ Zeros at − 1, 2, and 3 having degree 4 ◮ Zeros at 2 3 and − 3 4 having degree 2
Behavior near an x -intercept An important application of calculus concepts is to approximate complicated expressions with simpler ones. To visualize what a polynomial looks like near an x -intercept is easy: For example, let’s try to visualize the previous function p ( x ) = 1 2 ( x − 1) 3 (2 x + 3) 2 ( 1 2 x + 2) near its intercept at (1 , 0). To do this, we simply let x = 1 in all factors of p ( x ) except for the factor ( x − 1) that corresponds to the zero x = 1. This will give us the new function f ( x ) = 1 2 ( x − 1) 3 (2 + 3) 2 ( 1 2 + 2), which can be rewritten f ( x ) = 125 4 ( x − 1) 3
Behavior near an x -intercept We can see that this is correct by graphing p ( x ) and f ( x ) on the same axis: p ( x ) = 1 2 ( x − 1) 3 (2 x + 3) 2 ( 1 2 x + 2) and f ( x ) = 125 4 ( x − 1) 3
Behavior toward infinity An important application of calculus concepts is to approximate complicated expressions with simpler ones. To visualize what a polynomial looks like as x gets very far away from 0 (toward negative infinity or positive infinity) is also easy: For example, let’s try to visualize the behavior of the function p ( x ) = 1 2 ( x − 1) 3 (2 x + 3) 2 ( 1 2 x + 2) as x gets far away from 0. Each individual factor is easy to analyze: When x is huge, x − 1 behaves like x , 2 x + 3 behaves like 2 x , and 1 2 x + 2 behaves like 1 2 x . So as x gets far away from 0, we will have 1 2( x − 1) 3 (2 x + 3) 2 (1 2 x + 2) ∼ 1 2( x ) 3 (2 x ) 2 (1 2 x ) = x 6 So p ( x ) behaves like the function g ( x ) = x 6 for values of x far from 0. In other words, p ( x ) acts like its highest degree term for these values of x .
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