Learning Objectives

# Problem 1 of 3

f(x) = x^{3} + 3x^{2} - x - 3

(a) Find all the intercepts of f.

(b) Sketch the graph of f.

(c) Given x^{3} + 3x^{2} - x - 3 ≥ 0, write the solution set using interval notation.

**Solution**

**STEP I:**

First, you have to find the possible numerators, possible denominators, and possible outcomes. You find the possible numerators from the coeifficent with the highest degree, the factors of the coeifficent, in this case it is ±1. Your possible denominators come from the coeifficent with the lowest degree, in this case ±3 and ±1.

**Your possible outcomes are:**

±1/1 or ±3/1

**STEP II:**

now use synthesis division on steroids to find the intercepts.

-now that we have at least one intercept, we could use the divisions by synthesis to find the equation:

the numbers are the coeifficents to the equation the has the factor (x-1)

now we can generate a equation and use the factor theorem to find the other intercepts:

**SOLUTION A:**

(x-1)(x²+4x+3)

(x-1)(x+1)(x+3)

**The x-intercepts are: x=-1, x=1, and x=-3**

**The y-intercept is at -3**.

To find the y-intercept, you sub in 0 for where it is x.

**EXAMPLE:**

f(x)=x³+3x²-x-3

f(0)=(0)³+3(0)²-(0)-3

f(0)=-3

STEP III:

use sign analysis to see where the the parobola dip or bounces.

**SOLUTION B:**

STEPIV:

draw the graph now that you know where all intercepts are and where the parabola dips or bounces.

and there is your graph :D

**SOLUTION C:**

this is how the graph looks like when the equation is x^{3} + 3x^{2} - x - 3 ≥ 0

**{x|x >1; x € R}**

# Problem 2 of 3

Given

(a) Find all the intercepts of g.

(b) Identify the vertical and horizontal asymptotes if any.

(c) Sketch the graph.

**Solution**

When solving rational equations, there are some steps that we follow:

1)Factor

2)Y-int

3)Roots

4)Vertical Asymptote

5)Horizontal Asymptote

6)Sign test

7)Graph

**Lets Begin**

# Problem 3 of 3

coming soon ...

Solution

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