Find A Polynomial Of Degree 3 With Real Coefficients And Zeros Of \[$-3\$\], \[$-1\$\], And \[$4\$\], For Which \[$f(-2) = -30\$\].$\[ F(x) = \square \\](Simplify Your Answer. Do Not Factor.)
Understanding the Problem
When dealing with polynomials, it's essential to understand the relationship between the coefficients and the zeros of the polynomial. In this case, we're given three zeros: -3, -1, and 4, and we need to find a polynomial of degree 3 with real coefficients that satisfies these conditions.
The Factorized Form of a Polynomial
A polynomial with real coefficients can be written in the factorized form as:
f(x) = a(x - r1)(x - r2)(x - r3)
where a is a constant, and r1, r2, and r3 are the zeros of the polynomial.
Substituting the Given Zeros
We're given the zeros -3, -1, and 4, so we can substitute these values into the factorized form:
f(x) = a(x + 3)(x + 1)(x - 4)
Determining the Constant a
To find the value of a, we need to use the given condition f(-2) = -30. We can substitute x = -2 into the polynomial:
f(-2) = a(-2 + 3)(-2 + 1)(-2 - 4) f(-2) = a(1)(-1)(-6) f(-2) = 6a
Solving for a
Now we can solve for a by equating the expression to the given value -30:
6a = -30 a = -30/6 a = -5
The Final Polynomial
Now that we have the value of a, we can substitute it back into the factorized form to get the final polynomial:
f(x) = -5(x + 3)(x + 1)(x - 4)
Expanding the Polynomial
To get the polynomial in the standard form, we can expand the expression:
f(x) = -5(x^2 + 4x + 3)(x - 4) f(x) = -5(x^3 - 4x^2 + 4x^2 - 16x + 3x - 12) f(x) = -5(x^3 - 13x - 12)
The Final Answer
The final polynomial is:
f(x) = -5x^3 + 65x + 60
Conclusion
In this problem, we found a polynomial of degree 3 with real coefficients and zeros of -3, -1, and 4, for which f(-2) = -30. The final polynomial is -5x^3 + 65x + 60.
Understanding the Importance of Real Coefficients
When dealing with polynomials, it's essential to understand the relationship between the coefficients and the zeros of the polynomial. In this case, we used the fact that the polynomial has real coefficients to find the value of a.
The Role of Zeros in Polynomial Factorization
The zeros of a polynomial play a crucial role in its factorization. In this case, we used the given zeros to write the polynomial in the factorized form.
The Significance of the Constant a
The constant a is a crucial part of the polynomial factorization. In this case, we used the given condition f(-2) = -30 to find the value of a.
The Importance of Expanding the Polynomial
Expanding the polynomial is essential to get it in the standard form. In this case, we expanded the expression to get the final polynomial.
Real-World Applications of Polynomial Factorization
Polynomial factorization has numerous real-world applications, including engineering, physics, and computer science. In these fields, polynomial factorization is used to model and analyze complex systems.
Conclusion
In this article, we found a polynomial of degree 3 with real coefficients and zeros of -3, -1, and 4, for which f(-2) = -30. The final polynomial is -5x^3 + 65x + 60. We also discussed the importance of real coefficients, the role of zeros in polynomial factorization, the significance of the constant a, and the importance of expanding the polynomial.
Frequently Asked Questions
Q: What is the relationship between the coefficients and the zeros of a polynomial?
A: The coefficients of a polynomial are related to its zeros through the factorized form. In this case, we used the factorized form to find the polynomial with real coefficients and zeros of -3, -1, and 4.
Q: How do you find the value of a in the polynomial factorization?
A: To find the value of a, we need to use the given condition f(-2) = -30. We can substitute x = -2 into the polynomial and solve for a.
Q: What is the significance of the constant a in polynomial factorization?
A: The constant a is a crucial part of the polynomial factorization. It determines the scale of the polynomial, and its value affects the zeros of the polynomial.
Q: How do you expand a polynomial in the factorized form?
A: To expand a polynomial in the factorized form, we need to multiply the factors together. In this case, we expanded the expression to get the final polynomial.
Q: What are some real-world applications of polynomial factorization?
A: Polynomial factorization has numerous real-world applications, including engineering, physics, and computer science. In these fields, polynomial factorization is used to model and analyze complex systems.
Q: Can you provide an example of a polynomial with real coefficients and zeros of -3, -1, and 4?
A: Yes, the polynomial we found in this article is:
f(x) = -5x^3 + 65x + 60
This polynomial has real coefficients and zeros of -3, -1, and 4.
Q: How do you determine the degree of a polynomial?
A: The degree of a polynomial is determined by the highest power of the variable in the polynomial. In this case, the polynomial has a degree of 3.
Q: What is the difference between a polynomial with real coefficients and a polynomial with complex coefficients?
A: A polynomial with real coefficients has all real zeros, while a polynomial with complex coefficients has complex zeros.
Q: Can you provide a step-by-step guide to finding a polynomial with real coefficients and zeros of -3, -1, and 4?
A: Yes, here is a step-by-step guide:
- Write the polynomial in the factorized form: f(x) = a(x + 3)(x + 1)(x - 4)
- Substitute x = -2 into the polynomial and solve for a: 6a = -30
- Solve for a: a = -30/6 = -5
- Substitute the value of a back into the factorized form: f(x) = -5(x + 3)(x + 1)(x - 4)
- Expand the polynomial: f(x) = -5(x^3 - 13x - 12)
Q: What are some common mistakes to avoid when finding a polynomial with real coefficients and zeros?
A: Some common mistakes to avoid include:
- Not using the correct factorized form
- Not substituting the correct value of x into the polynomial
- Not solving for a correctly
- Not expanding the polynomial correctly
Conclusion
In this Q&A article, we answered some frequently asked questions about polynomial factorization, including the relationship between coefficients and zeros, the significance of the constant a, and the importance of expanding the polynomial. We also provided a step-by-step guide to finding a polynomial with real coefficients and zeros of -3, -1, and 4.