Write A Polynomial Function In Standard Form With Zeros At -4, 1, And 1.Use The Symbol ^ To Signify An Exponent. For Example, For 2 To The Third Power, Enter 2^3.y =
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 written with the term having the highest degree first, followed by the terms with lower degrees. In this article, we will learn how to write a polynomial function in standard form with zeros at -4, 1, and 1.
Zeros of a Polynomial Function
The zeros of a polynomial function are the values of the variable 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 can be found using various methods, including factoring, synthetic division, and the quadratic formula.
Writing a Polynomial Function with Given Zeros
To write a polynomial function with given zeros, we can use the fact that if a is a zero of the function, then (x - a) is a factor of the function. Therefore, if we have zeros at -4, 1, and 1, we can write the function as:
f(x) = a(x + 4)(x - 1)^2
where a is a constant coefficient.
Finding the Constant Coefficient
To find the constant coefficient a, we can use the fact that the product of the zeros of a polynomial function is equal to the constant term of the function, divided by the leading coefficient. In this case, the product of the zeros is (-4)(1)(1) = -4. Therefore, we can set up the equation:
a(-4)(1)(1) = -4
Solving for a, we get:
a = 1
Writing the Polynomial Function in Standard Form
Now that we have found the constant coefficient a, we can write the polynomial function in standard form:
f(x) = (x + 4)(x - 1)^2
To expand this expression, we can use the distributive property:
f(x) = (x + 4)(x^2 - 2x + 1)
Expanding further, we get:
f(x) = x^3 - 2x^2 + x + 4x^2 - 8x + 4
Combining like terms, we get:
f(x) = x^3 + 2x^2 - 7x + 4
Conclusion
Frequently Asked Questions
In this article, we will answer some frequently asked questions about writing polynomial functions in standard form.
Q: What is the standard form of a polynomial function?
A: The standard form of a polynomial function is written with the term having the highest degree first, followed by the terms with lower degrees. For example, the standard form of a polynomial function with zeros at -4, 1, and 1 is:
f(x) = x^3 + 2x^2 - 7x + 4
Q: How do I find the zeros of a polynomial function?
A: The zeros of a polynomial function are the values of the variable that make the function equal to zero. You can find the zeros of a polynomial function by factoring, synthetic division, or the quadratic formula.
Q: What is the relationship between the zeros of a polynomial function and its factors?
A: If a is a zero of the function, then (x - a) is a factor of the function. This means that if you know the zeros of a polynomial function, you can write the function as a product of its factors.
Q: How do I find the constant coefficient of a polynomial function?
A: To find the constant coefficient of a polynomial function, you can use the fact that the product of the zeros of a polynomial function is equal to the constant term of the function, divided by the leading coefficient.
Q: What is the difference between a polynomial function and a rational function?
A: A polynomial function is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. A rational function, on the other hand, is a mathematical expression consisting of variables and coefficients combined using addition, subtraction, multiplication, and division.
Q: Can I write a polynomial function in standard form if I don't know its zeros?
A: Yes, you can write a polynomial function in standard form even if you don't know its zeros. You can use the fact that the sum of the coefficients of a polynomial function is equal to the constant term of the function, divided by the leading coefficient.
Q: How do I expand a polynomial function in standard form?
A: To expand a polynomial function in standard form, you can use the distributive property. This involves multiplying each term of the function by each other term and combining like terms.
Q: What are some common mistakes to avoid when writing polynomial functions in standard form?
A: Some common mistakes to avoid when writing polynomial functions in standard form include:
- Not using the correct order of terms
- Not combining like terms
- Not using the distributive property to expand the function
- Not checking for errors in the function
Conclusion
In this article, we answered some frequently asked questions about writing polynomial functions in standard form. We covered topics such as the standard form of a polynomial function, finding the zeros of a polynomial function, and expanding a polynomial function in standard form. We also discussed some common mistakes to avoid when writing polynomial functions in standard form.