The Function Below Has At Least One Rational Zero. Use This Fact To Find All Zeros Of The Function.$ H(x) = 2x^3 + 5x^2 - 5x + 7 $If There Is More Than One Zero, Separate Them With Commas.
Introduction
In mathematics, the concept of rational zeros is a crucial aspect of polynomial functions. A rational zero is a value of x that makes the polynomial function equal to zero. In this article, we will explore the function h(x) = 2x^3 + 5x^2 - 5x + 7 and use the fact that it has at least one rational zero to find all zeros of the function.
Understanding Rational Zeros
Rational zeros are values of x that make the polynomial function equal to zero. To find rational zeros, we can use the Rational Root Theorem, which states that if a rational number p/q is a root of the polynomial function f(x), then p must be a factor of the constant term and q must be a factor of the leading coefficient.
Applying the Rational Root Theorem
To find the rational zeros of the function h(x) = 2x^3 + 5x^2 - 5x + 7, we can apply the Rational Root Theorem. The constant term is 7, and the leading coefficient is 2. Therefore, the possible rational zeros are ±1, ±7, ±1/2, and ±7/2.
Using Synthetic Division
To find the rational zeros, we can use synthetic division. Synthetic division is a method of dividing a polynomial function by a linear factor. We can use synthetic division to divide the polynomial function h(x) = 2x^3 + 5x^2 - 5x + 7 by each of the possible rational zeros.
Synthetic Division for x = 1
Let's start by dividing the polynomial function h(x) = 2x^3 + 5x^2 - 5x + 7 by x = 1.
| 2 | 5 | -5 | 7 |
|---|---|---|---|
| 1 | 7 | 2 | 0 |
The result of the synthetic division is 2x^2 + 7x + 2. This means that x = 1 is a rational zero of the polynomial function h(x) = 2x^3 + 5x^2 - 5x + 7.
Synthetic Division for x = -1
Next, let's divide the polynomial function h(x) = 2x^3 + 5x^2 - 5x + 7 by x = -1.
| 2 | 5 | -5 | 7 |
|---|---|---|---|
| -1 | -2 | 6 | 0 |
The result of the synthetic division is 2x^2 + 3x + 7. This means that x = -1 is not a rational zero of the polynomial function h(x) = 2x^3 + 5x^2 - 5x + 7.
Synthetic Division for x = 1/2
Next, let's divide the polynomial function h(x) = 2x^3 + 5x^2 - 5x + 7 by x = 1/2.
| 2 | 5 | -5 | 7 |
|---|---|---|---|
| 1/2 | 13/2 | -13/2 | 0 |
The result of the synthetic division is 2x^2 + 13x - 13/2. This means that x = 1/2 is not a rational zero of the polynomial function h(x) = 2x^3 + 5x^2 - 5x + 7.
Synthetic Division for x = -1/2
Next, let's divide the polynomial function h(x) = 2x^3 + 5x^2 - 5x + 7 by x = -1/2.
| 2 | 5 | -5 | 7 |
|---|---|---|---|
| -1/2 | -13/2 | 13/2 | 0 |
The result of the synthetic division is 2x^2 - 13x + 13/2. This means that x = -1/2 is not a rational zero of the polynomial function h(x) = 2x^3 + 5x^2 - 5x + 7.
Synthetic Division for x = 7
Next, let's divide the polynomial function h(x) = 2x^3 + 5x^2 - 5x + 7 by x = 7.
| 2 | 5 | -5 | 7 |
|---|---|---|---|
| 7 | 0 | 0 | 0 |
The result of the synthetic division is 0. This means that x = 7 is a rational zero of the polynomial function h(x) = 2x^3 + 5x^2 - 5x + 7.
Synthetic Division for x = -7
Next, let's divide the polynomial function h(x) = 2x^3 + 5x^2 - 5x + 7 by x = -7.
| 2 | 5 | -5 | 7 |
|---|---|---|---|
| -7 | 0 | 0 | 0 |
The result of the synthetic division is 0. This means that x = -7 is a rational zero of the polynomial function h(x) = 2x^3 + 5x^2 - 5x + 7.
Conclusion
In conclusion, we have found that the function h(x) = 2x^3 + 5x^2 - 5x + 7 has at least one rational zero, x = 1. We have also found that x = 7 and x = -7 are rational zeros of the function. Therefore, the zeros of the function are x = 1, x = 7, and x = -7.
Final Answer
Introduction
In our previous article, we explored the concept of rational zeros and applied the Rational Root Theorem to find the rational zeros of the polynomial function h(x) = 2x^3 + 5x^2 - 5x + 7. In this article, we will answer some frequently asked questions about rational zeros and provide additional examples to help solidify your understanding.
Q: What is the Rational Root Theorem?
A: The Rational Root Theorem states that if a rational number p/q is a root of the polynomial function f(x), then p must be a factor of the constant term and q must be a factor of the leading coefficient.
Q: How do I apply the Rational Root Theorem?
A: To apply the Rational Root Theorem, you need to identify the factors of the constant term and the leading coefficient. Then, you can use these factors to create a list of possible rational roots. You can use synthetic division to test each possible rational root and determine if it is indeed a root of the polynomial function.
Q: What is synthetic division?
A: Synthetic division is a method of dividing a polynomial function by a linear factor. It is a shortcut for long division and can be used to find the roots of a polynomial function.
Q: How do I use synthetic division to find the roots of a polynomial function?
A: To use synthetic division to find the roots of a polynomial function, you need to divide the polynomial function by each possible rational root. If the result of the synthetic division is 0, then the rational root is indeed a root of the polynomial function.
Q: What are some common mistakes to avoid when using the Rational Root Theorem?
A: Some common mistakes to avoid when using the Rational Root Theorem include:
- Not factoring the constant term and leading coefficient correctly
- Not creating a complete list of possible rational roots
- Not using synthetic division to test each possible rational root
- Not checking for repeated roots
Q: Can I use the Rational Root Theorem to find the roots of a polynomial function with complex coefficients?
A: No, the Rational Root Theorem only applies to polynomial functions with rational coefficients. If you have a polynomial function with complex coefficients, you will need to use a different method to find its roots.
Q: Can I use the Rational Root Theorem to find the roots of a polynomial function with a degree greater than 3?
A: Yes, the Rational Root Theorem can be used to find the roots of a polynomial function with a degree greater than 3. However, you will need to use synthetic division multiple times to find all the roots.
Q: What are some real-world applications of the Rational Root Theorem?
A: The Rational Root Theorem has many real-world applications, including:
- Finding the roots of polynomial equations that model real-world phenomena
- Determining the stability of a system
- Optimizing a function
Conclusion
In conclusion, the Rational Root Theorem is a powerful tool for finding the roots of polynomial functions. By understanding how to apply the theorem and using synthetic division, you can find the roots of even the most complex polynomial functions. Remember to avoid common mistakes and to use the theorem in conjunction with other methods to find the roots of polynomial functions.
Final Answer
The final answer is that the Rational Root Theorem is a useful tool for finding the roots of polynomial functions, but it has its limitations and should be used in conjunction with other methods to find the roots of polynomial functions.