SLIDE 1 Section 5.1
Fall 2013
SLIDE 2 Polynomial Functions
A polynomial is a sum of monomials. A monomial is an algebraic expression of the form c · xn, 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:
3
- 2. g(z) = πz10
- 3. h(x) = 5x2+1
6
2t2 + t +
√ 5
t2
2x2 + 2x1/2
- 3. h(x) = √x.
- 4. p(z) = z2+z
z
SLIDE 3
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(x2 + 5)
◮ What is the degree of this polynomial? ◮ What is the domain of this function? ◮ Find the x- and y-intercepts of f .
SLIDE 4
- 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(2x + 3)2(1 2x + 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.
SLIDE 5
Multiplicities
p(x) = 1
2(x − 1)3(2x + 3)2( 1 2x + 2)
SLIDE 6
Multiplicities
p(x) = 1
2(x − 1)3(2x + 3)2( 1 2x + 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 (2x + 3) factor tells us the graph
touches the x-axis at x = −3/2.
◮ The odd multiplicity of the ( 1 2x + 2) factor tells us the graph
crosses the x-axis at x = −4.
SLIDE 7
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
SLIDE 8
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(2x + 3)2( 1 2x + 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
SLIDE 9
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(2x + 3)2( 1 2x + 2) and f (x) = 125 4 (x − 1)3
SLIDE 10
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(2x + 3)2( 1 2x + 2) as x gets far away from 0.
Each individual factor is easy to analyze: When x is huge, x − 1 behaves like x, 2x + 3 behaves like 2x, and 1
2x + 2 behaves like 1 2x.
So as x gets far away from 0, we will have 1 2(x − 1)3(2x + 3)2(1 2x + 2) ∼ 1 2(x)3(2x)2(1 2x) = x6 So p(x) behaves like the function g(x) = x6 for values of x far from 0. In other words, p(x) acts like its highest degree term for these values of x.
