Consider the following function.
g(x) = 6x^4 − 13x^3 − 144x^2 + 325x − 150
(a) Use the zero or root feature of a graphing utility to approximate the zeros of the function accurate to three decimal places. (Enter your answers as a comma-separated list. Round your answers to three decimal places.)
x =
(b) Determine one of the exact zeros (use synthetic division to verify your result).
x =
(c) Factor the polynomial completely.
g(x) =
10mm=1n
To approximate the zeros of the function g(x) = 6x^4 − 13x^3 − 144x^2 + 325x − 150, we can use a graphing utility with a zero or root feature. Here's how you can do it:
(a) Using a Graphing Utility:
1. Open a graphing utility software or use an online graphing tool.
2. Enter the function g(x) = 6x^4 − 13x^3 − 144x^2 + 325x − 150.
3. Plot the graph of the function.
4. Look for points on the x-axis where the graph intersects or touches the x-axis. These are the approximate zeros of the function.
5. Use the zero or root feature of the graphing utility to find the x-values of these points accurate to three decimal places.
The approximated zeros of the function g(x) are represented as x = [approximate values].
(b) To determine one of the exact zeros, we can use synthetic division. Here's how:
1. Write down the function g(x) = 6x^4 − 13x^3 − 144x^2 + 325x − 150.
2. Choose a guess for a zero of the function. This guess is usually a factor of the constant term divided by a factor of the leading coefficient.
In this case, the constant term is -150, and the leading coefficient is 6. A possible guess for a zero is ± 1, ± 2, ± 3, ± 5, ± 6, ± 10, ± 15, ± 25, ± 30, ± 50, ± 75, ± 150.
3. Synthetic division:
- Choose a guess for a zero (let's say x = a).
- Set up the synthetic division table, with the coefficients of the function in descending order.
- Perform the synthetic division using the chosen guess.
- Check the remainder. If it is zero, then a is a zero of the function.
- If the remainder is not zero, repeat the process with another guess until a zero is found.
By performing synthetic division with different guessed zeros, you can find an exact zero of the function.
(c) To factor the polynomial completely, we first need to find the zeros of the function and then write the polynomial as a product of linear factors. Once we have the zeros, we can use them to write the factored form of the polynomial.
Using the approximate or exact zeros obtained from parts (a) and (b), we can factor the polynomial g(x).
g(x) = (x − [zero1])(x − [zero2])(x − [zero3])(x − [zero4])
Replace [zero1], [zero2], [zero3], [zero4] with the zeros you found in part (a) or (b).
The factored form of the polynomial g(x) will be g(x) = (x − [zero1])(x − [zero2])(x − [zero3])(x − [zero4]).
wolframalpha is your friend:
http://www.wolframalpha.com/input/?i=6x^4+%E2%88%92+13x^3+%E2%88%92+144x^2+%2B+325x+%E2%88%92+150