Write A Polynomial Function In Standard Form With Zeros At { -7, 0$}$, And ${ 1\$} .A. { F(x) = X(x+7)(x-1)$}$ B. { F(x) = X^3 - 6x^2 - 7x$}$ C. { F(x) = X^3 + 6x^2 - 7x$}$ D. [$f(x) =
Understanding Polynomial Functions
A polynomial function is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The standard form of a polynomial function is a way of expressing the function in a specific format, which is essential for various mathematical operations and applications.
Zeros of a Polynomial Function
The zeros of a polynomial function are the values of x that make the function equal to zero. In other words, they are the solutions to the equation f(x) = 0. The zeros of a polynomial function are crucial in understanding the behavior of the function and can be used to determine the function's graph.
Given Zeros and the Polynomial Function
We are given three zeros: -7, 0, and 1. We need to write a polynomial function in standard form that has these zeros.
Step 1: Determine the Factors
Since the zeros are -7, 0, and 1, we can write the factors of the polynomial function as (x + 7), x, and (x - 1).
Step 2: Multiply the Factors
To write the polynomial function in standard form, we need to multiply the factors together. We can start by multiplying the first two factors: (x + 7)x = x^2 + 7x.
Step 3: Multiply the Result with the Third Factor
Now, we multiply the result from Step 2 with the third factor: (x^2 + 7x)(x - 1) = x^3 - x^2 + 7x^2 - 7x = x^3 + 6x^2 - 7x.
Conclusion
Based on the given zeros, we have determined that the polynomial function in standard form is f(x) = x^3 + 6x^2 - 7x.
Answer
The correct answer is C. f(x) = x^3 + 6x^2 - 7x.
Why is this the correct answer?
This is the correct answer because the polynomial function f(x) = x^3 + 6x^2 - 7x has the given zeros: -7, 0, and 1. When we substitute x = -7, 0, or 1 into the function, we get f(-7) = 0, f(0) = 0, and f(1) = 0, respectively.
Comparison with Other Options
Let's compare our answer with the other options:
- Option A: f(x) = x(x + 7)(x - 1) = x^3 + 6x^2 - 7x. This is the same as our answer, but it is not in standard form.
- Option B: f(x) = x^3 - 6x^2 - 7x. This is not the correct answer because it does not have the given zeros.
- Option D: f(x) = x^3 + 6x^2 + 7x. This is not the correct answer because it does not have the given zeros.
Conclusion
In conclusion, the correct answer is C. f(x) = x^3 + 6x^2 - 7x. This polynomial function has the given zeros: -7, 0, and 1, and it is in standard form.
Key Takeaways
- A polynomial function is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
- The standard form of a polynomial function is a way of expressing the function in a specific format.
- The zeros of a polynomial function are the values of x that make the function equal to zero.
- To write a polynomial function in standard form, we need to multiply the factors together.
- The correct answer is C. f(x) = x^3 + 6x^2 - 7x.
References
- [1] "Polynomial Functions." Math Open Reference, mathopenref.com/polynomial.html.
- [2] "Standard Form of a Polynomial Function." Math Is Fun, mathisfun.com/algebra/polynomial-standard-form.html.
Additional Resources
- Khan Academy: Polynomial Functions
- Mathway: Polynomial Functions
- Wolfram Alpha: Polynomial Functions
Q&A: Writing Polynomial Functions in Standard Form =====================================================
Frequently Asked Questions
Q: What is a polynomial function?
A polynomial function is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
Q: What is the standard form of a polynomial function?
The standard form of a polynomial function is a way of expressing the function in a specific format, where the terms are arranged in descending order of the exponent of the variable.
Q: How do I determine the factors of a polynomial function?
To determine the factors of a polynomial function, you need to identify the zeros of the function. The zeros are the values of x that make the function equal to zero.
Q: How do I multiply the factors of a polynomial function?
To multiply the factors of a polynomial function, you need to multiply each factor together. You can start by multiplying the first two factors, and then multiply the result with the third factor.
Q: What is the correct answer for the given zeros: -7, 0, and 1?
The correct answer is C. f(x) = x^3 + 6x^2 - 7x.
Q: Why is this the correct answer?
This is the correct answer because the polynomial function f(x) = x^3 + 6x^2 - 7x has the given zeros: -7, 0, and 1. When we substitute x = -7, 0, or 1 into the function, we get f(-7) = 0, f(0) = 0, and f(1) = 0, respectively.
Q: How do I compare my answer with the other options?
To compare your answer with the other options, you need to check if the other options have the same zeros as your answer. If they do, then they are also correct.
Q: What are some common mistakes to avoid when writing polynomial functions in standard form?
Some common mistakes to avoid when writing polynomial functions in standard form include:
- Not identifying the zeros of the function
- Not multiplying the factors together correctly
- Not arranging the terms in descending order of the exponent of the variable
Q: How can I practice writing polynomial functions in standard form?
You can practice writing polynomial functions in standard form by using online resources such as Khan Academy, Mathway, or Wolfram Alpha. You can also try solving problems from a textbook or worksheet.
Q: What are some real-world applications of polynomial functions?
Some real-world applications of polynomial functions include:
- Modeling population growth
- Analyzing data
- Solving optimization problems
Conclusion
In conclusion, writing polynomial functions in standard form is an essential skill in mathematics. By understanding the concept of polynomial functions and how to write them in standard form, you can solve a wide range of problems and apply mathematical concepts to real-world situations.
Key Takeaways
- A polynomial function is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
- The standard form of a polynomial function is a way of expressing the function in a specific format.
- The zeros of a polynomial function are the values of x that make the function equal to zero.
- To write a polynomial function in standard form, you need to multiply the factors together.
- The correct answer is C. f(x) = x^3 + 6x^2 - 7x.
References
- [1] "Polynomial Functions." Math Open Reference, mathopenref.com/polynomial.html.
- [2] "Standard Form of a Polynomial Function." Math Is Fun, mathisfun.com/algebra/polynomial-standard-form.html.
Additional Resources
- Khan Academy: Polynomial Functions
- Mathway: Polynomial Functions
- Wolfram Alpha: Polynomial Functions