802.sagews

A polynomial function is something in the form:

$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0$

The degree is $n$.  The leading coefficient is $a_n$.

EXAMPLE 1  Using the Leading Coefficient Test

Determine the end behavior of $f(x) = x^3 + 3x^2 - x - 3$.

f(x) = x^3 + 3*x^2 - x - 3

plot(f, -5, 5)

$n$ is odd, and $a_n$ is positive.  Notice the end behavior.

EXAMPLE 2  Using the Leading Coefficient Test

Determine the end behavior of $f(x) = -4x^3(x - 1)^2(x + 5)$.

f(x) = -4*x^3*(x - 1)^2*(x + 5)
plot(f, -6, 6)

$n$ is even, and $a_n$ is negative.  Notice the end behavior.

EXAMPLE 3  Using the Leading Coefficient Test

Determine the end behavior of $f(x) = -49x^3 + 806x^2 + 3776x + 2503$

f(x) = -49*x^3 + 806*x^2 + 3776*x + 2503
plot(f, -100, 100)

$n$ is odd, and $a_n$ is negative.  Notice the end behavior.

EXAMPLE 4  Using the Leading Coefficient Test

Is this the following a complete graph of $f(x) = -x^4 + 8x^3 + 4x^2 + 2$?

f(x) = -x^4 + 8*x^3 + 4*x^2 + 2
plot(f, xmin = -8, xmax = 8, ymin = -10, ymax = 10)

EXAMPLE 5  Finding Zeros of a Polynomial Function

Find all zeros of $f(x) = x^3 + 3x^2 - x - 3$.

f(x) = x^3 + 3*x^2 - x - 3

factor(f)

solve(f, x)

plot(f, -4, 2)

EXAMPLE 6  Finding Zeros of a Polynomial Function

Find all zeros of $f(x) = -x^4 + 4x^3 - 4x^2$.

f(x) = -x^4 + 4*x^3 - 4*x^2

factor(f)

solve(f, x)

plot(f, -1, 3)

EXAMPLE 7  Finding Zeros and Their Multiplicities

Find the zeros of $f(x) = \frac{1}{2}(x + 1)(2x - 3)^2$ and their multiplicities.

f(x) = 1/2*(x + 1)*(2*x - 3)^2

solve(f == 0, x)

plot(f, -2, 2)

EXAMPLE 8  Using the Intermediate Value Theorem

Show that $f(x) = x^3 - 2x - 5$ has a real zero between 2 and 3.

f(x) = x^3 - 2*x - 5

f(2)

f(3)

plot(f, 1, 4)

solve(f == 0, x)

x1 = -1/2*(I*sqrt(3) + 1)*(1/18*sqrt(3)*sqrt(643) + 5/2)^(1/3) + 1/3*(I*sqrt(3) - 1)/(1/18*sqrt(3)*sqrt(643) + 5/2)^(1/3)

x2 = -1/2*(-I*sqrt(3) + 1)*(1/18*sqrt(3)*sqrt(643) + 5/2)^(1/3) + 1/3*(-I*sqrt(3) - 1)/(1/18*sqrt(3)*sqrt(643) + 5/2)^(1/3)

x3 = (1/18*sqrt(3)*sqrt(643) + 5/2)^(1/3) + 2/3/(1/18*sqrt(3)*sqrt(643) + 5/2)^(1/3)

x1.n()

x2.n()

x3.n()

EXAMPLE 9

Graphing a Polynomial Function

Graph:  $f(x) = x^4 - 2x^2 + 1$

f(x) = x^4 - 2*x^2 + 1

# a_n is positive, and degree is even, so both ends up

factor(f)

solve(f == 0, x)

x_intercept_1, x_intercept_2 = (-1, 0), (1, 0)

y_intercept = (0, f(0))

plot(f, -2, 2) + points([x_intercept_1, x_intercept_2, y_intercept], color = 'red')