Use the algorithm for curve sketching to sketch the graph of each function.

a) f(x) 4x^3+6x^2-24x-2

1. First to find intercepts

y=0

0=4x^3+6x^2-24x-2

=2(2x^3+3x^3-12x-1)

I don't know how to find the x-intercept. I can't use quadratic formula or synthetic division :S

ok, so you would have to solve

4x^3 + 6x^2 - 24x - 2 = 0

I also could not find any "nice" f(x) = 0
(I tried x = ±2, ±1/2)

so I went to the reliable Wolfram to get
http://www.wolframalpha.com/input/?i=4x%5E3+%2B+6x%5E2+-+24x+-+2+%3D+0

notice that all 3 roots appear to be irrational
making it a messy solution.

I checked in the book answer and the site u gave me matched with the answer.

How did the website get solutions as:

x= -3.28183
x= -1.863

There is no simple formula to solve a general cubic equation.

There are several methods that can be used to find solutions ...
- Newton's Method and some type of iteration algorithms are the most popular

once you have one of the roots of the cubic, you can do long division or use synthetic division to reduce the cubic to a quadratic.
From there you can use the quadratic formula to find the other two roots
I don't know what method the Wolfram site uses.

To find the x-intercepts of the function f(x) = 4x^3 + 6x^2 - 24x - 2, you can use numerical methods such as graphing or iterative methods. Here's an explanation of how to approach it using graphing:

1. Plot the function on a graphing calculator or software: Start by graphing the function f(x) = 4x^3 + 6x^2 - 24x - 2. This will give you an estimate of where the x-intercepts might be located.

2. Analyze the graph: Look at the graph and observe where it crosses the x-axis. These points will be the x-intercepts of the function. Take note of the approximate x-values.

3. Refine the approximation: If the graph only gives you a rough estimate of the x-intercepts, you can zoom in or change the scale to get a more accurate representation of the intercepts. This will help you determine the specific x-values where the graph crosses the x-axis.

4. Repeat if necessary: If the x-values obtained are still not precise enough, you can repeat the process with a higher level of zoom or use other graphing techniques to narrow down the x-intercepts further.

Note: The x-intercepts are the points where the graph of the function intersects the x-axis, meaning that the y-coordinate is zero (y = 0).

Remember that finding the exact x-intercepts of a polynomial function can be challenging, especially for higher-degree polynomials. Numerical approximations through graphing or iterative methods are often used to estimate the x-intercepts.