Solve: 2 X 3 − 50 X = 0 2x^3 - 50x = 0 2 X 3 − 50 X = 0 .A. 0 , 5 , − 5 0, 5, -5 0 , 5 , − 5 B. 0 , 5 I , − 5 I 0, 5i, -5i 0 , 5 I , − 5 I C. − 1 , 1 ± I 5 -1, 1 \pm I \sqrt{5} − 1 , 1 ± I 5 ​ D. 0 , 5 , 5 I 0, 5, 5i 0 , 5 , 5 I

Introduction

Solving polynomial equations is a fundamental concept in mathematics, and it is essential to understand the different techniques used to solve them. In this article, we will focus on solving the cubic equation $2x^3 - 50x = 0$. This equation can be solved using various methods, including factoring, the rational root theorem, and synthetic division. We will explore each of these methods and provide step-by-step solutions to the equation.

Factoring the Equation

The first step in solving the equation $2x^3 - 50x = 0$ is to factor out the greatest common factor (GCF) of the two terms. In this case, the GCF is $2x$. Factoring out $2x$ gives us:

$$2x(x^2 - 25) = 0$$

The Rational Root Theorem

The rational root theorem states that if a rational number $p/q$ is a root of the polynomial equation $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 = 0$, then $p$ must be a factor of the constant term $a_0$, and $q$ must be a factor of the leading coefficient $a_n$. In this case, the constant term is $0$, and the leading coefficient is $2$. Therefore, the possible rational roots of the equation are $\pm 1, \pm 5, \pm 10, \pm 25$.

Synthetic Division

Synthetic division is a method used to divide a polynomial by a linear factor. In this case, we can use synthetic division to divide the polynomial $x^2 - 25$ by the linear factor $(x - 5)$. The result of the synthetic division is:

$$\begin{array}{c|rrrr} 5 & 1 & 0 & -25 \\ & & 5 & 25 \\ \hline & 1 & 5 & 0 \end{array}$$

Solving the Equation

Now that we have factored the equation and used synthetic division to divide the polynomial, we can solve the equation. We have:

$$2x(x^2 - 25) = 0$$

This equation is true if either $2x = 0$ or $x^2 - 25 = 0$. Solving for $x$ in each case, we get:

$$2x = 0 \Rightarrow x = 0$$

$$x^2 - 25 = 0 \Rightarrow x^2 = 25 \Rightarrow x = \pm 5$$

Conclusion

In conclusion, the solutions to the equation $2x^3 - 50x = 0$ are $x = 0, 5, -5$. These solutions can be verified by plugging them back into the original equation.

Final Answer

The final answer is: $\boxed{0, 5, -5}$

Introduction

In our previous article, we solved the cubic equation $2x^3 - 50x = 0$ using various methods, including factoring, the rational root theorem, and synthetic division. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on solving polynomial equations.

Q: What is the greatest common factor (GCF) of the two terms in the equation $2x^3 - 50x = 0$?

A: The GCF of the two terms is $2x$. Factoring out $2x$ gives us $2x(x^2 - 25) = 0$.

Q: What is the rational root theorem, and how is it used to solve polynomial equations?

A: The rational root theorem states that if a rational number $p/q$ is a root of the polynomial equation $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 = 0$, then $p$ must be a factor of the constant term $a_0$, and $q$ must be a factor of the leading coefficient $a_n$. In this case, the constant term is $0$, and the leading coefficient is $2$. Therefore, the possible rational roots of the equation are $\pm 1, \pm 5, \pm 10, \pm 25$.

Q: What is synthetic division, and how is it used to solve polynomial equations?

A: Synthetic division is a method used to divide a polynomial by a linear factor. In this case, we can use synthetic division to divide the polynomial $x^2 - 25$ by the linear factor $(x - 5)$. The result of the synthetic division is:

$$\begin{array}{c|rrrr} 5 & 1 & 0 & -25 \\ & & 5 & 25 \\ \hline & 1 & 5 & 0 \end{array}$$

Q: How do I solve the equation $2x^3 - 50x = 0$?

A: To solve the equation, we need to find the values of $x$ that make the equation true. We can do this by factoring the equation and using synthetic division to divide the polynomial. The solutions to the equation are $x = 0, 5, -5$.

Q: What are some common mistakes to avoid when solving polynomial equations?

A: Some common mistakes to avoid when solving polynomial equations include:

• Not factoring the equation correctly
• Not using the rational root theorem to find possible rational roots
• Not using synthetic division to divide the polynomial
• Not checking the solutions to the equation to make sure they are correct

Q: How do I check the solutions to the equation to make sure they are correct?

A: To check the solutions to the equation, we need to plug them back into the original equation and make sure they make the equation true. In this case, we can plug $x = 0, 5, -5$ back into the equation $2x^3 - 50x = 0$ and make sure they make the equation true.

Conclusion

In conclusion, solving polynomial equations can be a challenging task, but with the right techniques and tools, it can be done. We hope this Q&A section has helped clarify any doubts and provided additional information on solving polynomial equations.

Final Answer

The final answer is: $\boxed{0, 5, -5}$