Expanding the Cubic Expression

Oct 13, 2025 by windailys.com 107 views

Introduction

In algebra, expanding a cubic expression involves multiplying out the terms to simplify the expression. In this article, we will focus on expanding the cubic expression $\left(y+2x^3\right)^3$ . This involves applying the binomial theorem, which is a powerful tool for expanding expressions of the form $(a+b)^n$ . We will break down the process into manageable steps, making it easier to understand and apply.

The Binomial Theorem

The binomial theorem states that for any positive integer $n$ , the expansion of $(a+b)^n$ is given by:

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

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

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

Expanding the Cubic Expression

To expand the cubic expression $\left(y+2x^3\right)^3$ , we will apply the binomial theorem with $a=y$ , $b=2x^3$ , and $n=3$ . This gives us:

\left(y+2x^3\right)^3 = \binom{3}{0}y^3 + \binom{3}{1}y^2(2x^3) + \binom{3}{2}y(2x^3)^2 + \binom{3}{3}(2x^3)^3

Calculating the Binomial Coefficients

Using the formula for the binomial coefficient, we can calculate the values of $\binom{3}{0}$ , $\binom{3}{1}$ , $\binom{3}{2}$ , and $\binom{3}{3}$ :

\binom{3}{0} = \frac{3!}{0!(3-0)!} = 1

\binom{3}{1} = \frac{3!}{1!(3-1)!} = 3

\binom{3}{2} = \frac{3!}{2!(3-2)!} = 3

\binom{3}{3} = \frac{3!}{3!(3-3)!} = 1

Substituting the Binomial Coefficients

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

\left(y+2x^3\right)^3 = 1y^3 + 3y^2(2x^3) + 3y(2x^3)^2 + 1(2x^3)^3

Simplifying the Expression

We can simplify the expression by evaluating the products:

\left(y+2x^3\right)^3 = y^3 + 6y^2x^3 + 12yx^6 + 8x^9

Conclusion

In this article, we have expanded the cubic expression $\left(y+2x^3\right)^3$ using the binomial theorem. We have calculated the binomial coefficients and substituted them into the expression, simplifying it to obtain the final result. This process demonstrates the power of the binomial theorem in expanding expressions of the form $(a+b)^n$ .

Answer

The correct answer is:

$y^3 + 6y^2x^3 + 12yx^6 + 8x^9$

Introduction

In our previous article, we expanded the cubic expression $\left(y+2x^3\right)^3$ using the binomial theorem. In this article, we will provide a Q&A guide to help you understand the process and apply it to similar problems.

Q: What is the binomial theorem?

A: The binomial theorem is a powerful tool for expanding expressions of the form $(a+b)^n$ . It states that for any positive integer $n$ , the expansion of $(a+b)^n$ is given by:

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

Q: How do I apply the binomial theorem?

A: To apply the binomial theorem, you need to identify the values of $a$ , $b$ , and $n$ in the expression you want to expand. Then, you can use the formula to calculate the binomial coefficients and substitute them into the expression.

Q: What are binomial coefficients?

A: Binomial coefficients are the numbers that appear in the binomial theorem. They are defined as:

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

Q: How do I calculate binomial coefficients?

A: To calculate binomial coefficients, you can use the formula:

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

For example, to calculate $\binom{3}{1}$ , you would use:

\binom{3}{1} = \frac{3!}{1!(3-1)!} = 3

Q: What is the difference between $\binom{n}{k}$ and $\binom{n}{n-k}$ ?

A: The binomial coefficients $\binom{n}{k}$ and $\binom{n}{n-k}$ are equal. This is because:

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

Q: Can I use the binomial theorem to expand expressions with negative exponents?

A: No, the binomial theorem is only applicable to expressions with non-negative exponents. If you have an expression with negative exponents, you will need to use a different method to expand it.

Q: Can I use the binomial theorem to expand expressions with fractional exponents?

A: No, the binomial theorem is only applicable to expressions with integer exponents. If you have an expression with fractional exponents, you will need to use a different method to expand it.

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:

Forgetting to calculate the binomial coefficients
Making errors when calculating the binomial coefficients
Failing to substitute the binomial coefficients into the expression
Not simplifying the expression after expanding it

Conclusion

In this article, we have provided a Q&A guide to help you understand the binomial theorem and apply it to similar problems. We have covered topics such as the definition of the binomial theorem, how to apply it, and common mistakes to avoid. By following these guidelines, you should be able to expand expressions of the form $(a+b)^n$ with confidence.

Additional Resources

If you need additional help or resources, you can try the following:

Check out online resources such as Khan Academy or Mathway
Consult a textbook or online tutorial on algebra
Practice expanding expressions using the binomial theorem
Ask a teacher or tutor for help

Answer

The correct answers to the questions in this article are:

Q1: The binomial theorem is a powerful tool for expanding expressions of the form $(a+b)^n$ .
Q2: To apply the binomial theorem, you need to identify the values of $a$ , $b$ , and $n$ in the expression you want to expand.
Q3: Binomial coefficients are the numbers that appear in the binomial theorem.
Q4: To calculate binomial coefficients, you can use the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ .
Q5: The binomial coefficients $\binom{n}{k}$ and $\binom{n}{n-k}$ are equal.
Q6: No, the binomial theorem is only applicable to expressions with non-negative exponents.
Q7: No, the binomial theorem is only applicable to expressions with integer exponents.
Q8: Some common mistakes to avoid when using the binomial theorem include forgetting to calculate the binomial coefficients, making errors when calculating the binomial coefficients, failing to substitute the binomial coefficients into the expression, and not simplifying the expression after expanding it.