The problem consists of three parts: a. Divide the polynomial $x^3 + 2x^2 - x - 2$ by $(x-1)$. b. Solve the following two equations: i. $x^2 + 5x - 24 = 0$ ii. $2x^2 + 3x - 1 = 0$ c. Express the repeating decimal $0.\overline{14}$ in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

1. Problem Description

The problem consists of three parts:

a. Divide the polynomial $x^3 + 2x^2 - x - 2$ by $(x-1)$.

b. Solve the following two equations:

i. $x^2 + 5x - 24 = 0$

ii. $2x^2 + 3x - 1 = 0$

c. Express the repeating decimal $0.\overline{14}$ in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

2. Solution Steps

a. Polynomial division:

We use synthetic division or long division to divide $x^3 + 2x^2 - x - 2$ by $x-1$.

Using synthetic division:

1 | 1 2 -1 -2

| 1 3 2

|----------------

1 3 2 0

The quotient is $x^2 + 3x + 2$.

b. Solving the equations:

i. $x^2 + 5x - 24 = 0$

We can factor this quadratic equation as $(x+8)(x-3)=0$.

Therefore, $x+8 = 0$ or $x-3 = 0$, which gives us $x=-8$ or $x=3$.

ii. $2x^2 + 3x - 1 = 0$

We can use the quadratic formula to solve for $x$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a=2$, $b=3$, and $c=-1$.

$x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-1)}}{2(2)} = \frac{-3 \pm \sqrt{9+8}}{4} = \frac{-3 \pm \sqrt{17}}{4}$

Therefore, $x = \frac{-3 + \sqrt{17}}{4}$ or $x = \frac{-3 - \sqrt{17}}{4}$.

c. Expressing $0.\overline{14}$ as a fraction:

Let $x = 0.\overline{14} = 0.141414...$

Then $100x = 14.141414...$

Subtracting $x$ from $100x$ gives:

$100x - x = 14.141414... - 0.141414...$

$99x = 14$

$x = \frac{14}{99}$

3. Final Answer

a. $x^2 + 3x + 2$

b. i. $x = -8, x = 3$

ii. $x = \frac{-3 + \sqrt{17}}{4}, x = \frac{-3 - \sqrt{17}}{4}$

c. $\frac{14}{99}$