Given $f(x)=x^3+3x^2-13x-15$. For Which Value Of $x$ Is $f(x) \neq 0$?A. -5 B. -3 C. -1 D. 3
Introduction
In this article, we will explore the solution to a cubic equation, specifically the function f(x) = x^3 + 3x^2 - 13x - 15. We are tasked with finding the value of x for which f(x) is not equal to 0. This involves factoring the cubic equation, identifying its roots, and determining the value of x that satisfies the given condition.
Understanding the Cubic Equation
A cubic equation is a polynomial equation of degree three, which means the highest power of the variable (in this case, x) is three. The general form of a cubic equation is ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants. In our case, the cubic equation is f(x) = x^3 + 3x^2 - 13x - 15.
Factoring the Cubic Equation
To find the roots of the cubic equation, we need to factor it. Factoring involves expressing the cubic equation as a product of linear factors, which will help us identify the values of x that satisfy the equation. We can start by looking for common factors or using the rational root theorem to narrow down the possible values of x.
Using 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) + ... + a_1x + a_0 = 0, then p must be a factor of a_0, and q must be a factor of a_n. In our case, a_0 = -15, and a_n = 1. The factors of -15 are ±1, ±3, ±5, and ±15. Therefore, the possible rational roots of the cubic equation are ±1, ±3, ±5, and ±15.
Finding the Roots of the Cubic Equation
Using the rational root theorem, we can test the possible rational roots by substituting them into the cubic equation. We can start by testing the smaller values, such as ±1 and ±3. By substituting x = -3 into the cubic equation, we get:
f(-3) = (-3)^3 + 3(-3)^2 - 13(-3) - 15 = -27 + 27 + 39 - 15 = 24 ≠0
This tells us that x = -3 is not a root of the cubic equation. However, we can see that the expression inside the parentheses is equal to 24, which is not equal to 0. This suggests that x = -3 is a value of x for which f(x) ≠0.
Conclusion
In conclusion, we have found that x = -3 is a value of x for which f(x) ≠0. This is because when we substitute x = -3 into the cubic equation, we get a non-zero result. Therefore, the correct answer is B. -3.
Additional Discussion
It's worth noting that the cubic equation f(x) = x^3 + 3x^2 - 13x - 15 has three roots, which are the values of x that satisfy the equation. The other two roots are x = -1 and x = 3, which can be found using other methods such as synthetic division or the quadratic formula. However, for the purpose of this discussion, we are only interested in finding the value of x for which f(x) ≠0.
Final Answer
The final answer is B. -3.
Introduction
In our previous article, we explored the solution to a cubic equation, specifically the function f(x) = x^3 + 3x^2 - 13x - 15. We found that x = -3 is a value of x for which f(x) ≠0. In this article, we will answer some frequently asked questions related to the cubic equation and its solution.
Q: What is a cubic equation?
A: A cubic equation is a polynomial equation of degree three, which means the highest power of the variable (in this case, x) is three. The general form of a cubic equation is ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants.
Q: How do I factor a cubic equation?
A: Factoring a cubic equation involves expressing it as a product of linear factors. This can be done using various methods, such as the rational root theorem, synthetic division, or the quadratic formula. However, factoring a cubic equation can be challenging, and it may require some trial and error.
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 equation a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0 = 0, then p must be a factor of a_0, and q must be a factor of a_n. This theorem can help us narrow down the possible values of x that satisfy the equation.
Q: How do I find the roots of a cubic equation?
A: Finding the roots of a cubic equation involves substituting possible values of x into the equation and checking if the result is equal to 0. We can also use methods such as synthetic division or the quadratic formula to find the roots.
Q: What is the difference between a root and a solution?
A: A root of an equation is a value of x that satisfies the equation, while a solution is a value of x that satisfies the equation and also satisfies any additional conditions or constraints.
Q: Can I use a calculator to solve a cubic equation?
A: Yes, you can use a calculator to solve a cubic equation. Many calculators have built-in functions for solving polynomial equations, including cubic equations.
Q: How do I know if a value of x is a root of the equation?
A: To determine if a value of x is a root of the equation, substitute the value into the equation and check if the result is equal to 0. If the result is not equal to 0, then the value of x is not a root of the equation.
Q: Can I use the quadratic formula to solve a cubic equation?
A: No, the quadratic formula is used to solve quadratic equations, not cubic equations. However, you can use the quadratic formula to solve a quadratic factor of a cubic equation.
Q: How do I graph a cubic equation?
A: To graph a cubic equation, use a graphing calculator or a computer algebra system. You can also use a graphing tool such as Desmos or GeoGebra to visualize the graph of the equation.
Conclusion
In conclusion, solving a cubic equation involves factoring the equation, finding the roots, and determining the value of x that satisfies the equation. We hope that this Q&A article has provided you with a better understanding of the cubic equation and its solution.
Additional Resources
For more information on solving cubic equations, we recommend the following resources:
- Khan Academy: Solving Cubic Equations
- Mathway: Solving Cubic Equations
- Wolfram Alpha: Solving Cubic Equations
Final Answer
The final answer is B. -3.