(a) verify the given factors of f(x) (b) find the remaining...

(a) verify the given factors of $f(x)$ (b) find the remaining factor(s) of $f(x)$ (c) use your results to write the complete factorization of $f(x),$ (d) list all real zeros of $f,$ and (e) confirm your results by using a graphing utility to graph the function. $$\begin{array}{ll}\text { Function } & \text { Factors } \\f(x)=x^{3}+3 x^{2}-48 x-144 & (x+4 \sqrt{3}),(x+3)\end{array}$$

(a) verify the given factors of $f(x)$
(b) find the remaining factor(s) of $f(x)$
(c) use your results to write the complete factorization of $f(x),$
(d) list all real zeros of $f,$ and
(e) confirm your results by using a graphing utility to graph the function.
$$\begin{array}{ll}\text { Function } & \text { Factors } \\f(x)=x^{3}+3 x^{2}-48 x-144 & (x+4 \sqrt{3}),(x+3)\end{array}$$

Key Concepts

Factor Theorem

The Factor Theorem is a fundamental concept in algebra that states if a polynomial f(x) has a factor (x - c), then f(c) = 0. It is used to verify whether a proposed linear factor actually divides the polynomial without remainder, providing a straightforward method to test potential factors.

Polynomial Division

Polynomial division, including synthetic division and long division, is a method for dividing a polynomial by a linear (or sometimes higher degree) divisor. It is essential for breaking down a polynomial into simpler factors, helping to isolate and identify remaining factors once some factors have been verified.

Complete Factorization

Complete factorization involves expressing a polynomial as a product of its irreducible factors, typically linear or quadratic factors that cannot be further factored over the reals. This process simplifies the polynomial into its constituent parts, revealing all roots and simplifying further analysis.

Zeros of a Polynomial

Finding the zeros of a polynomial involves determining the values of x for which the polynomial equals zero. These zeros correspond to the solutions of the equation f(x) = 0 and are directly related to the factors of the polynomial, with each factor (x - c) indicating a zero at x = c.

Graphing Utility for Verification

A graphing utility is a tool that provides a visual representation of a polynomial function. By graphing the function, one can visually confirm the approximate locations of the zeros, check for consistency with algebraic factorization, and better understand the behavior of the polynomial across its domain.

Transcript

00:01 All right, so first i need to verify the given factors of my function.

00:04 So to do that, i'm going to use synthetic division.

00:07 So the factors that gives me are x plus 4 root 3 and x plus 3.

00:15 So since i'm using synthetic division, i need to use the root, meaning if i set this equal to 0 and solve, i would get x equals negative 3 and x equals negative 4 root 3.

00:27 So those would be the two values i'm dividing by doing my synthetic.

00:30 Division.

00:32 So i'll start with the negative three first.

00:34 It doesn't matter which one you start with, though.

00:36 And then i'm going to write down my coefficients.

00:38 So i'll have one, three, negative 48, and negative 144.

00:46 Drop the first term.

00:48 Multiply 1 times negative 3 is negative 3.

00:53 Add them up, and you get 0.

00:55 0 times negative 3 is 0.

00:57 Add them up, that's negative 48, negative 4 .5.

01:00 Negative and negative 3 is 144, add them up and you get 0.