Expand The Function.${ F(x) = (x-3)^4 }$ { = \, X^4 + \square X^3 + \square X^2 + \square X + \square \}

Introduction

In mathematics, expanding a function is a crucial step in understanding its behavior and properties. The given function, $f(x) = (x-3)^4$, is a polynomial function that can be expanded to reveal its coefficients and terms. In this article, we will walk you through the process of expanding this function, and provide a step-by-step guide to understanding the algebraic expression.

The Binomial Theorem

The binomial theorem is a mathematical formula that allows us to expand expressions of the form $(a+b)^n$, where $a$ and $b$ are numbers or variables, and $n$ is a positive integer. The theorem states that:

$$(a+b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \ldots + \binom{n}{n-1} a^1 b^{n-1} + \binom{n}{n} a^0 b^n$$

where $\binom{n}{k}$ is the binomial coefficient, defined as:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Expanding the Function

Now that we have the binomial theorem, we can apply it to expand the function $f(x) = (x-3)^4$. We will use the formula:

$$(x-3)^4 = \binom{4}{0} x^4 (-3)^0 + \binom{4}{1} x^3 (-3)^1 + \binom{4}{2} x^2 (-3)^2 + \binom{4}{3} x^1 (-3)^3 + \binom{4}{4} x^0 (-3)^4$$

Calculating the Binomial Coefficients

To calculate the binomial coefficients, we will use the formula:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

For $n=4$ and $k=0$, we have:

$$\binom{4}{0} = \frac{4!}{0!(4-0)!} = 1$$

For $n=4$ and $k=1$, we have:

$$\binom{4}{1} = \frac{4!}{1!(4-1)!} = 4$$

For $n=4$ and $k=2$, we have:

$$\binom{4}{2} = \frac{4!}{2!(4-2)!} = 6$$

For $n=4$ and $k=3$, we have:

$$\binom{4}{3} = \frac{4!}{3!(4-3)!} = 4$$

For $n=4$ and $k=4$, we have:

$$\binom{4}{4} = \frac{4!}{4!(4-4)!} = 1$$

Substituting the Binomial Coefficients

Now that we have calculated the binomial coefficients, we can substitute them into the formula:

$$(x-3)^4 = \binom{4}{0} x^4 (-3)^0 + \binom{4}{1} x^3 (-3)^1 + \binom{4}{2} x^2 (-3)^2 + \binom{4}{3} x^1 (-3)^3 + \binom{4}{4} x^0 (-3)^4$$

$$(x-3)^4 = 1 \cdot x^4 \cdot (-3)^0 + 4 \cdot x^3 \cdot (-3)^1 + 6 \cdot x^2 \cdot (-3)^2 + 4 \cdot x^1 \cdot (-3)^3 + 1 \cdot x^0 \cdot (-3)^4$$

$$(x-3)^4 = x^4 - 12x^3 + 54x^2 - 108x + 81$$

Conclusion

In this article, we have expanded the function $f(x) = (x-3)^4$ using the binomial theorem. We have calculated the binomial coefficients and substituted them into the formula to obtain the expanded expression. The final result is:

$$(x-3)^4 = x^4 - 12x^3 + 54x^2 - 108x + 81$$

This expanded expression reveals the coefficients and terms of the polynomial function, and provides a deeper understanding of its behavior and properties.

Further Reading

For further reading on the binomial theorem and its applications, we recommend the following resources:

• Binomial Theorem on Wikipedia
• Binomial Theorem on MathWorld
• Binomial Theorem on Khan Academy

Glossary

• Binomial Coefficient: A number that represents the number of ways to choose $k$ items from a set of $n$ items, without regard to order.
• Binomial Theorem: A mathematical formula that allows us to expand expressions of the form $(a+b)^n$, where $a$ and $b$ are numbers or variables, and $n$ is a positive integer.
• 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 $a_n, a_{n-1}, \ldots, a_1, a_0$ are constants, and $n$ is a positive integer.
  Q&A: Expanding the Function =============================

Q: What is the binomial theorem?

A: The binomial theorem is a mathematical formula that allows us to expand expressions of the form $(a+b)^n$, where $a$ and $b$ are numbers or variables, and $n$ is a positive integer.

Q: How do I apply the binomial theorem to expand a function?

A: To apply the binomial theorem, you need to follow these steps:

1. Identify the function you want to expand, which is in the form $(a+b)^n$.
2. Calculate the binomial coefficients using the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
3. Substitute the binomial coefficients into the formula $(a+b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \ldots + \binom{n}{n-1} a^1 b^{n-1} + \binom{n}{n} a^0 b^n$.
4. Simplify the expression to obtain the expanded form of the function.

Q: What are the binomial coefficients?

A: The binomial coefficients are numbers that represent the number of ways to choose $k$ items from a set of $n$ items, without regard to order. They are calculated using the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.

Q: How do I calculate the binomial coefficients?

A: To calculate the binomial coefficients, you need to use the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. For example, if you want to calculate the binomial coefficient $\binom{4}{2}$, you would use the formula:

$$\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{24}{2 \cdot 2} = 6$$

Q: What is the expanded form of the function $f(x) = (x-3)^4$?

A: The expanded form of the function $f(x) = (x-3)^4$ is:

$$(x-3)^4 = x^4 - 12x^3 + 54x^2 - 108x + 81$$

Q: How do I use the expanded form of the function?

A: The expanded form of the function can be used to:

• Simplify expressions involving the function
• Evaluate the function at specific values of $x$
• Use the function in algebraic manipulations

Q: What are some common applications of the binomial theorem?

A: The binomial theorem has many applications in mathematics, including:

• Expanding polynomial functions
• Evaluating expressions involving binomial coefficients
• Solving algebraic equations
• Calculating probabilities and statistics

Q: Where can I learn more about the binomial theorem?

A: You can learn more about the binomial theorem by:

• Reading online resources, such as Wikipedia or MathWorld
• Using online calculators or software, such as Wolfram Alpha or Mathematica
• Consulting textbooks or reference materials on algebra and mathematics
• Practicing problems and exercises to reinforce your understanding of the binomial theorem.