Example 7 from Algebra and Trigonometry, 5.2 Power Functions and Polynomial Functions

Identifying End Behavior and Degree of a Polynomial Function

Given the function $f (x) = - 3 x^{2} (x - 1) (x + 4),$ express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Obtain the general form by expanding the given expression for $f (x) .$

$\begin{matrix} f (x) & = & −3 x^{2} (x - 1) (x + 4) \\ = & −3 x^{2} (x^{2} + 3 x - 4) \\ = & −3 x^{4} - 9 x^{3} + 12 x^{2} \end{matrix}$
The general form is $f (x) = −3 x^{4} - 9 x^{3} + 12 x^{2} .$ The leading term is $−3 x^{4};$ therefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (–3), so the end behavior is
$\begin{array}{l} as x \to - \infty, f (x) \to - \infty \\ as x \to \infty, f (x) \to - \infty \end{array}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Given the function $f (x) = 0.2 (x - 2) (x + 1) (x - 5),$ express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.

How did it go?

Solution

Your turn

Solution

Your turn