Find All Zeros Of The Function $f(x) = 8x^3 - 26x^2 + 17x + 6$. Enter The Zeros Separated By Commas.

Feb 26, 2025 by ADMIN 101 views

**Finding the Zeros of a Cubic Function: A Step-by-Step Guide**

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Introduction

In mathematics, a zero of a function is a value of the variable that makes the function equal to zero. Finding the zeros of a function is an essential task in algebra and calculus, as it helps us understand the behavior of the function and its graph. In this article, we will focus on finding the zeros of a cubic function, specifically the function $f(x) = 8x^3 - 26x^2 + 17x + 6$ . We will use various methods to find the zeros of this function and provide the solutions in a comma-separated list.

What are Zeros of a Function?

Before we dive into finding the zeros of the given function, let's briefly discuss what zeros of a function are. A zero of a function is a value of the variable that makes the function equal to zero. In other words, if we have a function $f(x)$ , then a zero of the function is a value $x$ such that $f(x) = 0$ . Zeros of a function are also known as roots or solutions of the function.

Methods for Finding Zeros

There are several methods for finding the zeros of a function, including:

Factoring: This method involves expressing the function as a product of simpler functions, called factors, and then setting each factor equal to zero to find the zeros.
Synthetic Division: This method involves using a table to divide the function by a linear factor and find the remainder, which can be used to find the zeros.
Numerical Methods: These methods involve using numerical techniques, such as the Newton-Raphson method, to approximate the zeros of the function.

Finding the Zeros of the Given Function

In this section, we will use the factoring method to find the zeros of the given function $f(x) = 8x^3 - 26x^2 + 17x + 6$ . To do this, we need to factor the function into simpler factors.

Step 1: Factor the Function

To factor the function, we need to find two numbers whose product is $8 \times 6 = 48$ and whose sum is $-26$ . These numbers are $-24$ and $-1$ , so we can write the function as:

$f(x) = 8x^3 - 26x^2 + 17x + 6 = (8x^3 - 24x^2) + (-2x^2 + 17x) + 6$

Step 2: Factor the First Two Terms

The first two terms can be factored as:

$(8x^3 - 24x^2) = 8x^2(x - 3)$

So, the function becomes:

$f(x) = 8x^2(x - 3) + (-2x^2 + 17x) + 6$

Step 3: Factor the Last Two Terms

The last two terms can be factored as:

$(-2x^2 + 17x) = -x(2x - 17)$

So, the function becomes:

$f(x) = 8x^2(x - 3) - x(2x - 17) + 6$

Step 4: Factor the Function

Now, we can factor the function as:

$f(x) = (8x^2 - x)(x - 3) + 6$

Step 5: Find the Zeros

To find the zeros, we need to set each factor equal to zero and solve for $x$ . The first factor is $8x^2 - x$ , which can be factored as:

$8x^2 - x = x(8x - 1)$

So, we have:

$x(8x - 1) = 0$

This gives us two possible values for $x$ : $x = 0$ and $x = \frac{1}{8}$ .

The second factor is $x - 3$ , which gives us the value $x = 3$ .

Conclusion

In this article, we used the factoring method to find the zeros of the function $f(x) = 8x^3 - 26x^2 + 17x + 6$ . We factored the function into simpler factors and then set each factor equal to zero to find the zeros. The zeros of the function are $x = 0, x = \frac{1}{8},$ and $x = 3$ .

Final Answer

The zeros of the function $f(x) = 8x^3 - 26x^2 + 17x + 6$ are:

$0, \frac{1}{8}, 3$

Note: The zeros are separated by commas as requested.

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Q: What is a zero of a function?

A: A zero of a function is a value of the variable that makes the function equal to zero. In other words, if we have a function $f(x)$ , then a zero of the function is a value $x$ such that $f(x) = 0$ .

Q: Why is finding the zeros of a function important?

A: Finding the zeros of a function is important because it helps us understand the behavior of the function and its graph. Zeros of a function can also be used to solve equations and inequalities involving the function.

Q: What are some common methods for finding the zeros of a function?

A: Some common methods for finding the zeros of a function include:

Factoring: This method involves expressing the function as a product of simpler functions, called factors, and then setting each factor equal to zero to find the zeros.
Synthetic Division: This method involves using a table to divide the function by a linear factor and find the remainder, which can be used to find the zeros.
Numerical Methods: These methods involve using numerical techniques, such as the Newton-Raphson method, to approximate the zeros of the function.

Q: How do I know which method to use to find the zeros of a function?

A: The choice of method depends on the type of function and the complexity of the problem. For example, if the function can be easily factored, then factoring may be the best method to use. If the function is a polynomial of high degree, then numerical methods may be more suitable.

Q: Can I use a calculator to find the zeros of a function?

A: Yes, you can use a calculator to find the zeros of a function. Many calculators have built-in functions for finding the zeros of a polynomial, such as the "solve" function on a graphing calculator.

Q: How do I check my work when finding the zeros of a function?

A: To check your work, you can plug the zeros you found back into the original function to make sure that the function equals zero at those points. You can also use a graphing calculator to graph the function and verify that the zeros you found are indeed the points where the function crosses the x-axis.

Q: What if I get stuck while trying to find the zeros of a function?

A: If you get stuck while trying to find the zeros of a function, don't be afraid to ask for help. You can ask a teacher, tutor, or classmate for assistance, or you can try searching online for resources and tutorials.

Q: Can I find the zeros of a function that is not a polynomial?

A: Yes, you can find the zeros of a function that is not a polynomial. However, the methods you use may be different. For example, if the function is a rational function, you may need to use methods such as partial fraction decomposition to find the zeros.

Q: How do I apply the knowledge of finding zeros to real-world problems?

A: The knowledge of finding zeros can be applied to a wide range of real-world problems, such as:

Physics: Finding the zeros of a function can help you model the motion of an object and predict its behavior.
Engineering: Finding the zeros of a function can help you design and optimize systems, such as electrical circuits or mechanical systems.
Economics: Finding the zeros of a function can help you model economic systems and make predictions about future trends.

Conclusion

In this article, we have answered some of the most frequently asked questions about finding the zeros of a cubic function. We have discussed the importance of finding zeros, common methods for finding zeros, and how to apply the knowledge of finding zeros to real-world problems. We hope that this article has been helpful in answering your questions and providing you with a better understanding of this important topic.