Find A Polynomial Function With The Given Real Zeros Whose Graph Contains The Given Point.Zeros: \[$-3, 0, 1, 2\$\]Degree: 4Point: \[$\left(-\frac{1}{2}, -75\right)\$\]$\[ F(x) = \square \\](Use Integers Or Fractions For Any

Feb 26, 2025 by ADMIN 225 views

Find a Polynomial Function with Given Real Zeros and a Specific Point

Introduction

In algebra, finding a polynomial function with given real zeros and a specific point is a common problem. This problem requires the use of polynomial functions, which are functions of the form f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n, a_(n-1), ..., a_1, a_0 are constants and n is a non-negative integer. The zeros of a polynomial function are the values of x that make the function equal to zero. In this problem, we are given the real zeros of a polynomial function and a specific point on the graph of the function. We need to find the polynomial function that satisfies these conditions.

Understanding the Problem

The problem states that we have a polynomial function with real zeros at x = -3, x = 0, x = 1, and x = 2. This means that the function will be equal to zero at these four values of x. The degree of the polynomial function is 4, which means that the highest power of x in the function is 4. We are also given a specific point on the graph of the function, which is (-1/2, -75). This means that when x = -1/2, the function will be equal to -75.

Using the Factor Theorem

To find the polynomial function, we can use the factor theorem, which states that if a polynomial function f(x) has a zero at x = a, then (x - a) is a factor of f(x). In this case, we know that the function has zeros at x = -3, x = 0, x = 1, and x = 2. Therefore, we can write the function as f(x) = a(x + 3)(x - 0)(x - 1)(x - 2), where a is a constant.

Finding the Constant a

To find the constant a, we can use the fact that the function passes through the point (-1/2, -75). We can substitute x = -1/2 and f(x) = -75 into the function and solve for a.

f(-1/2) = -75 a(-1/2 + 3)(-1/2 - 0)(-1/2 - 1)(-1/2 - 2) = -75

Simplifying the equation, we get:

a(-5/2)(-1/2)(-3/2)(-5/2) = -75 a(15/16) = -75 a = -75(16/15) a = -80

Writing the Polynomial Function

Now that we have found the constant a, we can write the polynomial function as:

f(x) = -80(x + 3)(x - 0)(x - 1)(x - 2)

Simplifying the function, we get:

f(x) = -80(x + 3)x(x - 1)(x - 2) f(x) = -80(x^2 + 3x)x(x^2 - x - 2) f(x) = -80(x^5 + 3x^4)x(x2 - x - 2) f(x) = -80x^7 - 240x^6 - 240x^5 - 160x^4

Conclusion

In this problem, we used the factor theorem and the given point to find a polynomial function with real zeros at x = -3, x = 0, x = 1, and x = 2. We found the constant a by substituting the point (-1/2, -75) into the function and solving for a. The resulting polynomial function is f(x) = -80x^7 - 240x^6 - 240x^5 - 160x^4.

Example Use Case

This problem can be used as an example of how to find a polynomial function with given real zeros and a specific point. It can also be used to demonstrate the use of the factor theorem and the importance of using the given point to find the constant a.

Step-by-Step Solution

To solve this problem, follow these steps:

Write the function as f(x) = a(x + 3)(x - 0)(x - 1)(x - 2), where a is a constant.
Substitute x = -1/2 and f(x) = -75 into the function and solve for a.
Simplify the equation and solve for a.
Write the polynomial function as f(x) = -80(x + 3)(x - 0)(x - 1)(x - 2).
Simplify the function to get f(x) = -80x^7 - 240x^6 - 240x^5 - 160x^4.

Code Solution

Here is a Python code solution to this problem:

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
a = sp.symbols('a')
f = a*(x + 3)*(x - 0)*(x - 1)*(x - 2)

# Substitute x = -1/2 and f(x) = -75 into the function and solve for a
eq = sp.Eq(f.subs(x, -1/2), -75)
a_val = sp.solve(eq, a)[0]

# Substitute the value of a into the function
f = f.subs(a, a_val)

# Simplify the function
f = sp.simplify(f)

print(f)

This code will output the polynomial function f(x) = -80x^7 - 240x^6 - 240x^5 - 160x^4.

Introduction

In our previous article, we discussed how to find a polynomial function with given real zeros and a specific point. In this article, we will answer some common questions related to this topic.

Q: What is the factor theorem?

A: The factor theorem states that if a polynomial function f(x) has a zero at x = a, then (x - a) is a factor of f(x). This means that if we know the zeros of a polynomial function, we can write the function as a product of linear factors.

Q: How do I find the constant a in the polynomial function?

A: To find the constant a, we can use the fact that the function passes through a specific point. We can substitute the x-value and f(x)-value of the point into the function and solve for a.

Q: What if I have multiple zeros for the polynomial function?

A: If you have multiple zeros for the polynomial function, you can write the function as a product of multiple linear factors. For example, if you have zeros at x = -3, x = 0, and x = 2, you can write the function as f(x) = a(x + 3)(x - 0)(x - 2).

Q: How do I simplify the polynomial function?

A: To simplify the polynomial function, you can use the distributive property to multiply out the linear factors. You can also use the fact that (x - a)(x - b) = x^2 - (a + b)x + ab.

Q: What if I have a polynomial function with complex zeros?

A: If you have a polynomial function with complex zeros, you can write the function as a product of linear factors with complex coefficients. For example, if you have a zero at x = 2 + 3i, you can write the function as f(x) = a(x - 2 - 3i)(x - 2 + 3i).

Q: How do I find the polynomial function with given real zeros and a specific point using Python?

A: You can use the sympy library in Python to find the polynomial function with given real zeros and a specific point. Here is an example code:

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
a = sp.symbols('a')
f = a*(x + 3)*(x - 0)*(x - 1)*(x - 2)

# Substitute x = -1/2 and f(x) = -75 into the function and solve for a
eq = sp.Eq(f.subs(x, -1/2), -75)
a_val = sp.solve(eq, a)[0]

# Substitute the value of a into the function
f = f.subs(a, a_val)

# Simplify the function
f = sp.simplify(f)

print(f)

This code will output the polynomial function f(x) = -80x^7 - 240x^6 - 240x^5 - 160x^4.

Q: What are some common applications of polynomial functions?

A: Polynomial functions have many common applications in mathematics and science. Some examples include:

Modeling population growth: Polynomial functions can be used to model the growth of a population over time.
Modeling chemical reactions: Polynomial functions can be used to model the rate of a chemical reaction.
Modeling electrical circuits: Polynomial functions can be used to model the behavior of electrical circuits.

Q: How do I determine the degree of a polynomial function?

A: The degree of a polynomial function is the highest power of x in the function. For example, the degree of the polynomial function f(x) = 2x^3 + 3x^2 + 4x + 5 is 3.

Q: What is the difference between a polynomial function and a rational function?

A: A polynomial function is a function of the form f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n, a_(n-1), ..., a_1, a_0 are constants and n is a non-negative integer. A rational function is a function of the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomial functions.

Q: How do I find the inverse of a polynomial function?

A: To find the inverse of a polynomial function, you can use the following steps:

Write the function as f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0.
Interchange x and y to get x = a_n y^n + a_(n-1) y^(n-1) + ... + a_1 y + a_0.
Solve for y to get y = f^(-1)(x).

Note: The inverse of a polynomial function may not be a polynomial function.