Simplify The Expression:$f(x)=\left(3 X-2 X^2\right)^3$

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Introduction

In mathematics, simplifying expressions is a crucial step in solving problems and understanding complex concepts. The given expression, $f(x)=\left(3 x-2 x^2\right)^3$, is a polynomial raised to the power of 3. To simplify this expression, we need to apply the binomial theorem, which states that for any positive integer $n$, ${(a+b)^n = \binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1}b^1 + \binom{n}{2}a^{n-2}b^2 + \ldots + \binom{n}{n-1}a^1b^{n-1} + \binom{n}{n}a^0b^n}$.

Understanding the Binomial Theorem

The binomial theorem is a powerful tool for expanding expressions of the form $(a+b)^n$. It is based on the concept of combinations, which is denoted by the symbol $\binom{n}{k}$. The binomial theorem can be used to expand expressions with any number of terms, not just two. However, it is most commonly used for expressions with two terms, such as $(a+b)^n$.

Applying the Binomial Theorem

To simplify the given expression, we need to apply the binomial theorem. The expression $f(x)=\left(3 x-2 x^2\right)^3$ can be expanded using the binomial theorem as follows:

\begin{align*} f(x)&=\left(3 x-2 x^2\right)3\ &=\binom{3}{0}(3 x)^3(-2 x²⁾0 + \binom{3}{1}(3 x)^2(-2 x²⁾1 + \binom{3}{2}(3 x)^1(-2 x²⁾2 + \binom{3}{3}(3 x)^0(-2 x²⁾3\ &=27x^3-54x4+36x^5+8x6 \end{align*}

Explanation of the Simplification Process

The simplification process involves applying the binomial theorem to the given expression. The binomial theorem is used to expand the expression $(3 x-2 x^2)^3$. The expansion involves calculating the combinations of the terms in the expression and multiplying them together.

Step-by-Step Solution

To simplify the expression $f(x)=\left(3 x-2 x^2\right)^3$, we need to follow these steps:

1. Apply the binomial theorem to the expression $(3 x-2 x^2)^3$.
2. Calculate the combinations of the terms in the expression.
3. Multiply the terms together to get the expanded expression.

Conclusion

In conclusion, simplifying the expression $f(x)=\left(3 x-2 x^2\right)^3$ involves applying the binomial theorem. The binomial theorem is a powerful tool for expanding expressions of the form $(a+b)^n$. By applying the binomial theorem, we can simplify the given expression and get the expanded form.

Final Answer

The final answer to the problem is $f(x)=27x^3-54x^4+36x^5+8x^6$.

Frequently Asked Questions

Q: What is the binomial theorem?

A: The binomial theorem is a powerful tool for expanding expressions of the form $(a+b)^n$. It is based on the concept of combinations, which is denoted by the symbol $\binom{n}{k}$.

Q: How do I apply the binomial theorem?

A: To apply the binomial theorem, you need to calculate the combinations of the terms in the expression and multiply them together.

Q: What is the final answer to the problem?

A: The final answer to the problem is $f(x)=27x^3-54x^4+36x^5+8x^6$.

References

• Binomial Theorem
• Combinations
• Polynomial

Further Reading

• Simplifying Expressions
• Binomial Theorem Examples
• Polynomial Expansions
  

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Q&A: Simplifying Expressions with the Binomial Theorem

Q: What is the binomial theorem and how is it used?

A: The binomial theorem is a powerful tool for expanding expressions of the form $(a+b)^n$. It is based on the concept of combinations, which is denoted by the symbol $\binom{n}{k}$. The binomial theorem is used to expand expressions with any number of terms, not just two.

Q: How do I apply the binomial theorem to simplify an expression?

A: To apply the binomial theorem, you need to calculate the combinations of the terms in the expression and multiply them together. The formula for the binomial theorem is:

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

Q: What is the difference between a combination and a permutation?

A: A combination is a selection of items from a larger set, where the order of the items does not matter. A permutation is a selection of items from a larger set, where the order of the items does matter.

Q: How do I calculate the combinations in the binomial theorem?

A: To calculate the combinations in the binomial theorem, you need to use the formula:

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

where $n!$ is the factorial of $n$, $k!$ is the factorial of $k$, and $(n-k)!$ is the factorial of $n-k$.

Q: What is the final answer to the problem?

A: The final answer to the problem is $f(x)=27x^3-54x^4+36x^5+8x^6$.

Q: Can I use the binomial theorem to simplify expressions with negative numbers?

A: Yes, you can use the binomial theorem to simplify expressions with negative numbers. However, you need to be careful when working with negative numbers, as the binomial theorem may produce negative terms.

Q: Can I use the binomial theorem to simplify expressions with fractions?

A: Yes, you can use the binomial theorem to simplify expressions with fractions. However, you need to be careful when working with fractions, as the binomial theorem may produce fractions in the final answer.

Q: What are some common mistakes to avoid when using the binomial theorem?

A: Some common mistakes to avoid when using the binomial theorem include:

• Not calculating the combinations correctly
• Not multiplying the terms together correctly
• Not simplifying the final answer correctly

Q: How do I check my work when using the binomial theorem?

A: To check your work when using the binomial theorem, you need to:

• Calculate the combinations correctly
• Multiply the terms together correctly
• Simplify the final answer correctly

Conclusion

In conclusion, the binomial theorem is a powerful tool for expanding expressions of the form $(a+b)^n$. By understanding how to apply the binomial theorem, you can simplify complex expressions and solve problems with ease.

Final Answer

The final answer to the problem is $f(x)=27x^3-54x^4+36x^5+8x^6$.

Frequently Asked Questions

Q: What is the binomial theorem?

A: The binomial theorem is a powerful tool for expanding expressions of the form $(a+b)^n$. It is based on the concept of combinations, which is denoted by the symbol $\binom{n}{k}$.

Q: How do I apply the binomial theorem?

A: To apply the binomial theorem, you need to calculate the combinations of the terms in the expression and multiply them together.

Q: What is the final answer to the problem?

A: The final answer to the problem is $f(x)=27x^3-54x^4+36x^5+8x^6$.

References

• Binomial Theorem
• Combinations
• Polynomial

Further Reading

• Simplifying Expressions
• Binomial Theorem Examples
• Polynomial Expansions