Select The Correct Answer.If F ( X ) = 2 X 2 − 4 X − 3 F(x) = 2x^2 - 4x - 3 F ( X ) = 2 X 2 − 4 X − 3 And G ( X ) = ( X − 4 ) ( X + 4 G(x) = (x - 4)(x + 4 G ( X ) = ( X − 4 ) ( X + 4 ], Find F ( X ) − G ( X F(x) - G(x F ( X ) − G ( X ].A. 4 X + 13 4x + 13 4 X + 13 B. X 2 − 4 X + 13 X^2 - 4x + 13 X 2 − 4 X + 13 C. 3 X 2 − 4 X + 13 3x^2 - 4x + 13 3 X 2 − 4 X + 13 D. − 4 X 2 − 4 X + 13 -4x^2 - 4x + 13 − 4 X 2 − 4 X + 13

Introduction

In mathematics, polynomial functions play a crucial role in various fields, including algebra, calculus, and engineering. These functions are used to model real-world phenomena, such as population growth, chemical reactions, and electrical circuits. In this article, we will focus on solving polynomial functions, specifically the difference between two given functions, $f(x)$ and $g(x)$.

Understanding the Functions

The first function, $f(x) = 2x^2 - 4x - 3$, is a quadratic function, which means it has a degree of 2. This function can be graphed as a parabola, with its vertex at the point where the function changes from increasing to decreasing or vice versa.

The second function, $g(x) = (x - 4)(x + 4)$, is also a quadratic function, but it is expressed in factored form. This form makes it easier to identify the roots of the function, which are the values of $x$ that make the function equal to zero.

Finding the Difference

To find the difference between $f(x)$ and $g(x)$, we need to subtract the second function from the first function. This can be done by distributing the negative sign to the terms inside the parentheses of the second function.

$f(x) - g(x) = (2x^2 - 4x - 3) - ((x - 4)(x + 4))$

To simplify this expression, we need to expand the product inside the parentheses of the second function.

$(x - 4)(x + 4) = x^2 - 16$

Now, we can rewrite the difference as:

$f(x) - g(x) = (2x^2 - 4x - 3) - (x^2 - 16)$

Simplifying the Expression

To simplify the expression, we need to combine like terms. We can do this by adding or subtracting the coefficients of the same variables.

$f(x) - g(x) = 2x^2 - 4x - 3 - x^2 + 16$

Combining like terms, we get:

$f(x) - g(x) = x^2 - 4x + 13$

Conclusion

In this article, we have solved the difference between two polynomial functions, $f(x)$ and $g(x)$. We have shown that the difference is equal to $x^2 - 4x + 13$. This result can be verified by plugging in values of $x$ into both functions and checking if the difference is indeed equal to the result.

Answer

The correct answer is:

• B. $x^2 - 4x + 13$

Discussion

This problem requires a good understanding of polynomial functions, including their properties and operations. The student should be able to identify the degree of the functions, expand the product inside the parentheses, and combine like terms to simplify the expression.

Tips and Variations

• To make this problem more challenging, you can ask the student to find the difference between two more complex polynomial functions.
• You can also ask the student to graph the functions and identify the vertex of the parabola.
• To make this problem easier, you can provide the student with a simplified expression and ask them to verify the result by plugging in values of $x$ into both functions.

Related Topics

• Polynomial Functions: These are functions that can be written in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$, where $a_n \neq 0$.
• Quadratic Functions: These are polynomial functions of degree 2, which can be written in the form $f(x) = ax^2 + bx + c$, where $a \neq 0$.
• Factored Form: This is a way of writing a polynomial function as a product of two or more binomial factors.

Glossary

• Polynomial Function: A function that can be written in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$, where $a_n \neq 0$.
• Quadratic Function: A polynomial function of degree 2, which can be written in the form $f(x) = ax^2 + bx + c$, where $a \neq 0$.
• Factored Form: A way of writing a polynomial function as a product of two or more binomial factors.
  Frequently Asked Questions: Polynomial Functions =====================================================

Q: What is a polynomial function?

A: A polynomial function is a function that can be written in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$, where $a_n \neq 0$. Polynomial functions can be classified based on their degree, which is the highest power of the variable $x$.

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

A: A quadratic function is a polynomial function of degree 2, which can be written in the form $f(x) = ax^2 + bx + c$, where $a \neq 0$. Quadratic functions are a special type of polynomial function and have a parabolic shape when graphed.

Q: How do I find the difference between two polynomial functions?

A: To find the difference between two polynomial functions, you need to subtract the second function from the first function. This can be done by distributing the negative sign to the terms inside the parentheses of the second function.

Q: What is the factored form of a polynomial function?

A: The factored form of a polynomial function is a way of writing the function as a product of two or more binomial factors. This form makes it easier to identify the roots of the function, which are the values of $x$ that make the function equal to zero.

Q: How do I graph a polynomial function?

A: To graph a polynomial function, you need to identify the degree of the function and the direction of the graph. For example, a quadratic function with a positive leading coefficient will have a parabolic shape that opens upwards, while a quadratic function with a negative leading coefficient will have a parabolic shape that opens downwards.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the function changes from increasing to decreasing or vice versa. The vertex can be found using the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

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

A: To find the roots of a polynomial function, you need to set the function equal to zero and solve for $x$. This can be done using various methods, including factoring, the quadratic formula, and numerical methods.

Q: What is the significance of polynomial functions in real-world applications?

A: Polynomial functions are used to model real-world phenomena, such as population growth, chemical reactions, and electrical circuits. They are also used in various fields, including engineering, economics, and computer science.

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

A: To determine the degree of a polynomial function, you need to identify the highest power of the variable $x$ in the function. For example, the function $f(x) = 2x^3 + 3x^2 - 4x + 1$ has a degree of 3.

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

A: A rational function is a function that can be written in the form $f(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomial functions. Polynomial functions are a special type of rational function where the denominator is a constant.

Q: How do I simplify a polynomial function?

A: To simplify a polynomial function, you need to combine like terms and eliminate any common factors. This can be done using various methods, including factoring, the distributive property, and the commutative property.

Q: What is the importance of polynomial functions in algebra?

A: Polynomial functions are a fundamental concept in algebra and are used to model various types of relationships between variables. They are also used to solve equations and inequalities, and to graph functions.

Q: How do I use polynomial functions to solve equations and inequalities?

A: To use polynomial functions to solve equations and inequalities, you need to set the function equal to a specific value and solve for $x$. This can be done using various methods, including factoring, the quadratic formula, and numerical methods.

Q: What is the relationship between polynomial functions and other types of functions?

A: Polynomial functions are related to other types of functions, such as rational functions, exponential functions, and trigonometric functions. They can be used to model various types of relationships between variables and can be combined with other functions to create more complex functions.