Question

1. $e(x) = \frac{x - 2}{x^3 - 172x - 336}$

Explanation:

Step1: Factor the denominator

We need to factor the cubic polynomial \(x^{3}-172x - 336\). Let's try to find rational roots using the Rational Root Theorem. The possible rational roots are factors of \(336\) divided by factors of \(1\), so \(\pm1,\pm2,\pm3,\pm4,\pm6,\pm7,\pm8,\pm12,\pm14,\pm16,\pm21,\pm24,\pm28,\pm42,\pm48,\pm56,\pm84,\pm112,\pm168,\pm336\).

Let's test \(x = 14\): \(14^{3}-172\times14 - 336=2744-2408 - 336=2744-(2408 + 336)=2744 - 2744 = 0\). So \((x - 14)\) is a factor.

Now we perform polynomial division or use synthetic division to divide \(x^{3}-172x - 336\) by \((x - 14)\).

Using synthetic division:

$$
LATEXBLOCK0
$$

So \(x^{3}-172x - 336=(x - 14)(x^{2}+14x + 24)\). Now factor \(x^{2}+14x + 24\): we need two numbers that multiply to \(24\) and add to \(14\). Those numbers are \(12\) and \(2\). So \(x^{2}+14x + 24=(x + 12)(x + 2)\).

Thus, the denominator factors as \((x - 14)(x + 12)(x + 2)\).

Step2: Simplify the function

The function \(e(x)=\frac{x - 2}{(x - 14)(x + 12)(x + 2)}\). There are no common factors between the numerator \((x - 2)\) and the denominator \((x - 14)(x + 12)(x + 2)\), so the function is already in its simplified form (except for finding domain restrictions, but if the problem is just to simplify the rational function, this is the simplified form).

Answer:

\(e(x)=\frac{x - 2}{(x - 14)(x + 12)(x + 2)}\) (or if we consider factoring the denominator completely as above)