Question

Use the Binomial Theorem to write the binomial expansion. \(\left(3 u+v^2\right)^6\)

Short answer

The binomial expansion of \((3u+v^2)^6\) is \(729u^{6} + 4374u^{5}v^{2} + 10935u^{4}v^{4} + 10935u^{3}v^{6} + 4374u^{2}v^{8} + 729uv^{10} + v^{12}\)

Step-by-step solution

Identify the coefficients and components for the binomial expansion

First, identify that \(a = 3u\), \(b = v^2\) and \(n = 6\).

Use the Binomial Theorem

Use the Binomial Theorem for each term, and sum all the terms to write the expansion. The Binomial Theorem states \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\). So, we need to calculate \(\binom{6}{k} (3u)^{6-k} (v^2)^k\) for each \(k\) from 0 to 6. This can be simplified further by recognizing that \( (3u)^{6-k}\) can be written as \(729u^{6-k}\) and \( (v^2)^k\) is \(v^{2k}\).

Expand and Simplify

Calculate each term of the series and add them together. The coefficients \(\binom{6}{k}\) can be calculated as \(\frac{6!}{k!(6-k)!}\). So, the expansion is \(\binom{6}{0}729u^{6}v^{0} + \binom{6}{1}729u^{5}v^{2} + \binom{6}{2}729u^{4}v^{4} + \binom{6}{3}729u^{3}v^{6} + \binom{6}{4}729u^{2}v^{8} + \binom{6}{5}729u^{1}v^{10} + \binom{6}{0}729u^{0}v^{12}\). This simplifies to \(729u^{6} + 4374u^{5}v^{2} + 10935u^{4}v^{4} + 10935u^{3}v^{6} + 4374u^{2}v^{8} + 729uv^{10} + v^{12}\)

Key concepts

Binomial Expansion

The binomial expansion is a way to express the powers of a binomial expression using the Binomial Theorem. If you have a binomial expression like \((a + b)^n\), the Binomial Theorem provides a formula to expand it into a sum of terms. Each term includes a combinatorial coefficient, a power of \(a\), and a power of \(b\).
Here is what the Binomial Theorem states:

• \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\)
• \(\binom{n}{k}\) is known as a binomial coefficient.

This formula helps in expanding the binomial into a polynomial with multiple terms.
In practice, the theorem allows us to calculate each term separately by identifying two main components and using the formula to find their values.
This is particularly useful in algebra to simplify expressions and solve equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division. In the binomial expansion, we work with expressions involving variables raised to powers.
For example, in \((3u + v^2)^6\), we have:

• 3u is a monomial.
• v^2 is another monomial.

These two monomials combined under the operation of addition form the binomial expression.
Using the Binomial Theorem, this expression is expanded to form a polynomial where each term consists of a coefficient and a product of powers of \(3u\) and \(v^2\). This manipulation of algebraic expressions is key to solving complex algebra problems.

Combinatorial Coefficients

Combinatorial coefficients are used in the binomial expansion to determine the size of each term in the expansion. These coefficients are also known as binomial coefficients and are denoted by \(\binom{n}{k}\).
They represent the number of ways to choose \(k\) items from a set of \(n\) items, and are calculated using the formula:

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

In the problem \((3u + v^2)^6\), combinatorial coefficients like \(\binom{6}{0}\), \(\binom{6}{1}\), etc., are used for each term in the expansion.
The values of these coefficients determine the relative weight of each term in the polynomial expansion. They are essential for calculating each term accurately in any binomial expansion.

Powers and Exponents

Powers and exponents play a crucial role in the binomial expansion by defining the degree of each variable in the terms formed. In mathematical terms, a power refers to the number of times a number, or a variable, is multiplied by itself.
For the expression \((3u + v^2)^6\), each term of the expanded polynomial will include powers of \(3u\) and \(v^2\).
Key points include:

• \((3u)^{6-k}\) means \(3u\) raised to the power \(6-k\).
• \((v^2)^k\) simplifies to \(v^{2k}\) by multiplying the exponent of \(v\) with \(k\).

This results in terms like \(u^{6}\), \(u^{4}v^{4}\), or \(v^{12}\), each having specific exponents from the multiplication of terms.
Understanding how powers and exponents work is key to calculating each part of the polynomial accurately.