The First Polynomial Expansion

Oct 13, 2025 by wbrum.org 99 views

Introduction

In algebra, polynomial expansions are a crucial concept in understanding various mathematical operations. The expansion of a polynomial involves expressing it as a sum of simpler polynomials, known as terms. In this article, we will focus on finding the coefficients of specific terms in two different polynomial expansions. We will use the binomial theorem to expand the given polynomials and then identify the coefficients of the desired terms.

The First Polynomial Expansion

We are given the polynomial $$\left(2 x^2+\frac{k2}{x}\right)^5$$ and asked to find the coefficient of $$x^4$$ in its expansion. To do this, we will use the binomial theorem, which states that for any positive integer $$n$$, the expansion of $$(a+b)^n$$ is given by:

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

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

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

In our case, we have $$a = 2x^2$$ and $$b = \frac{k^2}{x}$$. We want to find the coefficient of $$x^4$$ in the expansion of $$(2x^2 + \frac{k^2}{x})5$$.

Using the binomial theorem, we can expand the given polynomial as:

\left(2 x^2+\frac{k^2}{x}\right)^5 = \sum_{k=0}^{5} \binom{5}{k} (2x^2)^{5-k} \left(\frac{k^2}{x}\right)^k

To find the coefficient of $$x^4$$, we need to find the term in the expansion that has $$x^4$$ as a factor. This means that the power of $$x$$ in the term must be 4. We can see that this will happen when $$k = 2$$, since:

\binom{5}{2} (2x^2)^{5-2} \left(\frac{k^2}{x}\right)^2 = 10 \cdot 32x^6 \cdot \frac{k^4}{x^2} = 320k^4x^4

Therefore, the coefficient of $$x^4$$ in the expansion of $$\left(2 x^2+\frac{k2}{x}\right)^5$$ is $$320k^4$$.

The Second Polynomial Expansion

We are given the polynomial $$(2kx - 1)^4$$ and asked to find the coefficient of $$x^2$$ in its expansion. To do this, we will again use the binomial theorem.

Using the binomial theorem, we can expand the given polynomial as:

(2kx - 1)^4 = \sum_{k=0}^{4} \binom{4}{k} (2kx)^{4-k} (-1)^k

To find the coefficient of $$x^2$$, we need to find the term in the expansion that has $$x^2$$ as a factor. This means that the power of $$x$$ in the term must be 2. We can see that this will happen when $$k = 2$$, since:

\binom{4}{2} (2kx)^{4-2} (-1)^2 = 6 \cdot 4k^2x^2 \cdot 1 = 24k^2x^2

Therefore, the coefficient of $$x^2$$ in the expansion of $$(2kx - 1)^4$$ is $$24k^2$$.

Conclusion

In this article, we used the binomial theorem to expand two different polynomials and find the coefficients of specific terms in their expansions. We found that the coefficient of $$x^4$$ in the expansion of $$\left(2 x^2+\frac{k2}{x}\right)^5$$ is $$320k^4$$, and the coefficient of $$x^2$$ in the expansion of $$(2kx - 1)^4$$ is $$24k^2$$. These results demonstrate the importance of the binomial theorem in algebraic expansions and provide a foundation for further mathematical exploration.

References

Binomial Theorem. (n.d.). In Encyclopedia of Mathematics. Retrieved from https://encyclopediaofmath.org/wiki/Binomial_theorem
Algebra. (n.d.). In Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra

Q: What is the binomial theorem?

A: The binomial theorem is a mathematical formula that describes the expansion of a binomial expression, which is an expression consisting of two terms. It is used to expand expressions of the form $$(a+b)^n$$, where $$a$$ and $$b$$ are constants and $$n$$ is a positive integer.

Q: How do I use the binomial theorem to expand a polynomial?

A: To use the binomial theorem to expand a polynomial, you need to identify the binomial expression and the power to which it is raised. Then, you can use the formula to expand the expression. For example, to expand $$(2x^2 + \frac{k^2}{x})5$$, you would use the formula:

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

where $$a = 2x^2$$, $$b = \frac{k^2}{x}$$, and $$n = 5$$.

Q: What is the coefficient of a term in a polynomial expansion?

A: The coefficient of a term in a polynomial expansion is the numerical value that multiplies the variable part of the term. For example, in the expansion of $$(2x^2 + \frac{k^2}{x})5$$, the coefficient of the term $$x^4$$ is $$320k^4$$.

Q: How do I find the coefficient of a specific term in a polynomial expansion?

A: To find the coefficient of a specific term in a polynomial expansion, you need to identify the term and then use the formula for the binomial theorem to find the coefficient. For example, to find the coefficient of $$x^2$$ in the expansion of $$(2kx - 1)^4$$, you would use the formula:

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

where $$a = 2kx$$, $$b = -1$$, and $$n = 4$$.

Q: What is the difference between a binomial and a polynomial?

A: A binomial is an expression consisting of two terms, while a polynomial is an expression consisting of multiple terms. For example, $$(2x^2 + \frac{k^2}{x})$$ is a binomial, while $$(2x^2 + \frac{k^2}{x})5$$ is a polynomial.

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

A: Yes, you can use the binomial theorem to expand polynomials with negative exponents. However, you need to be careful when dealing with negative exponents, as they can lead to complex expressions.

Q: Are there any limitations to the binomial theorem?

A: Yes, there are limitations to the binomial theorem. It only works for binomial expressions, and it only works for positive integer powers. If you have a polynomial with a non-integer power or a non-binomial expression, you will need to use a different method to expand it.

Q: Can I use the binomial theorem to find the roots of a polynomial?

A: No, the binomial theorem is not used to find the roots of a polynomial. Instead, it is used to expand polynomials and find the coefficients of specific terms. To find the roots of a polynomial, you will need to use a different method, such as factoring or the quadratic formula.

Q: Are there any real-world applications of the binomial theorem?

A: Yes, the binomial theorem has many real-world applications. It is used in a wide range of fields, including physics, engineering, economics, and computer science. Some examples of real-world applications of the binomial theorem include:

Modeling population growth and decay
Analyzing the behavior of complex systems
Optimizing business processes and supply chains
Developing algorithms for computer science and machine learning

Q: Can I use the binomial theorem to solve differential equations?

A: No, the binomial theorem is not used to solve differential equations. Instead, it is used to expand polynomials and find the coefficients of specific terms. To solve differential equations, you will need to use a different method, such as separation of variables or numerical methods.