How Many Zeros Does The Following Equation Have?$7x^3 + X^3 - 2x + 9x = 0$A. 2 B. 3 C. 4 D. 6

Feb 26, 2025 by ADMIN 99 views

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Introduction

In this article, we will explore the concept of zeros in a polynomial equation. A zero of a polynomial is a value of the variable that makes the polynomial equal to zero. We will examine the given equation $7x^3 + x^3 - 2x + 9x = 0$ and determine the number of zeros it has.

Understanding the Equation

The given equation is a cubic equation, which means it has a degree of 3. The equation can be simplified by combining like terms:

7x^3 + x^3 - 2x + 9x = 8x^3 + 7x = 0

Factoring Out Common Terms

We can factor out the common term $x$ from the equation:

x(8x^2 + 7) = 0

Finding the Zeros

To find the zeros of the equation, we need to set each factor equal to zero and solve for $x$ . The first factor is $x$ , which is equal to zero when $x = 0$ . The second factor is $8x^2 + 7$ , which is equal to zero when $8x^2 = -7$ . However, this equation has no real solutions, as the square of any real number is non-negative.

Conclusion

Based on our analysis, we can conclude that the equation $7x^3 + x^3 - 2x + 9x = 0$ has only one zero, which is $x = 0$ . This is because the second factor $8x^2 + 7$ has no real solutions.

Final Answer

The final answer is $\boxed{1}$ .

However, the question asks for the number of zeros, and the options given are A. 2, B. 3, C. 4, D. 6. Since we have only one zero, we need to consider the complex zeros of the equation.

Complex Zeros

The equation $8x^2 + 7 = 0$ has two complex zeros, which are given by the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, $a = 8$ , $b = 0$ , and $c = 7$ . Plugging these values into the formula, we get:

x = \frac{0 \pm \sqrt{0 - 4(8)(7)}}{2(8)}

x = \frac{0 \pm \sqrt{-224}}{16}

x = \frac{0 \pm 2i\sqrt{56}}{16}

x = \frac{0 \pm i\sqrt{14}}{4}

Total Number of Zeros

In addition to the real zero $x = 0$ , the equation has two complex zeros. Therefore, the total number of zeros is 3.

Final Answer

The final answer is $\boxed{3}$ .

Note

The question asks for the number of zeros, and the options given are A. 2, B. 3, C. 4, D. 6. However, the correct answer is B. 3, which includes the real zero and the two complex zeros.

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Introduction

In our previous article, we explored the concept of zeros in a polynomial equation and determined the number of zeros for the given equation $7x^3 + x^3 - 2x + 9x = 0$ . In this article, we will answer some frequently asked questions about zeros in polynomial equations.

Q&A

Q: What is a zero in a polynomial equation?

A: A zero of a polynomial is a value of the variable that makes the polynomial equal to zero. In other words, it is a solution to the equation.

Q: How do I find the zeros of a polynomial equation?

A: To find the zeros of a polynomial equation, you need to set the equation equal to zero and solve for the variable. You can use various methods such as factoring, the quadratic formula, or numerical methods.

Q: What is the difference between a real zero and a complex zero?

A: A real zero is a value of the variable that is a real number, whereas a complex zero is a value of the variable that is a complex number, which can be expressed in the form $a + bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: Can a polynomial equation have no zeros?

A: Yes, a polynomial equation can have no zeros. For example, the equation $x^2 + 1 = 0$ has no real zeros, but it has two complex zeros.

Q: Can a polynomial equation have multiple zeros?

A: Yes, a polynomial equation can have multiple zeros. For example, the equation $x(x - 1)(x - 2) = 0$ has three zeros: $x = 0$ , $x = 1$ , and $x = 2$ .

Q: How do I determine the number of zeros of a polynomial equation?

A: To determine the number of zeros of a polynomial equation, you need to examine the degree of the polynomial. The degree of a polynomial is the highest power of the variable. For example, the polynomial $x^3 + 2x^2 - 3x + 1$ has a degree of 3, which means it can have at most 3 zeros.

Q: Can a polynomial equation have a zero that is not a real number?

A: Yes, a polynomial equation can have a zero that is not a real number. For example, the equation $x^2 + 1 = 0$ has two complex zeros.

Q: How do I find the zeros of a polynomial equation with complex coefficients?

A: To find the zeros of a polynomial equation with complex coefficients, you can use various methods such as the quadratic formula or numerical methods. You can also use the fact that complex zeros come in conjugate pairs.

Conclusion

In this article, we have answered some frequently asked questions about zeros in polynomial equations. We have discussed the concept of zeros, how to find zeros, the difference between real and complex zeros, and how to determine the number of zeros of a polynomial equation.

Final Thoughts

Understanding zeros in polynomial equations is an important concept in mathematics, and it has many applications in science, engineering, and other fields. By mastering this concept, you can solve a wide range of problems and make new discoveries.

Note

The questions and answers in this article are based on the concept of zeros in polynomial equations. If you have any further questions or need more clarification, please feel free to ask.