Question

Expand each expression using the Binomial Theorem. $$ \left(x^{2}-y^{2}\right)^{6} $$

Short answer

The expansion is \[ x^{12} - 6x^{10}y^2 + 15x^8y^4 - 20x^6y^6 + 15x^4y^8 - 6x^2y^{10} + y^{12} \]

Step-by-step solution

- Understand the Binomial Theorem

The Binomial Theorem states that \[ (a+b)^n = \sum\_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] where \( \binom{n}{k} \) is a binomial coefficient. Here, we will use \( a = x^2 \), \( b = -y^2 \), and \( n = 6 \).

- Identify the Binomial Coefficients

First, identify the binomial coefficients \( \binom{6}{k} \) for each \( k \) from 0 to 6. They are:\[ \binom{6}{0}, \binom{6}{1}, \binom{6}{2}, \binom{6}{3}, \binom{6}{4}, \binom{6}{5}, \binom{6}{6} \].

- Calculate Each Binomial Coefficient

Calculate each binomial coefficient using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \):\[ \binom{6}{0} = 1, \binom{6}{1} = 6, \binom{6}{2} = 15, \binom{6}{3} = 20, \binom{6}{4} = 15, \binom{6}{5} = 6, \binom{6}{6} = 1 \].

- Substitute into Binomial Expansion

Substitute \( a = x^2 \), \( b = -y^2 \), and \( n = 6 \) into the binomial expansion formula and write it out:\[ (x^2 - y^2)^6 = \sum_{k=0}^{6} \binom{6}{k} (x^2)^{6-k} (-y^2)^k \].

- Expand for Each Term

Expand the summation for each term from \( k = 0 \) to \( k = 6 \):- For \( k = 0 \): \( \binom{6}{0} (x^2)^6 (-y^2)^0 = 1 \cdot x^{12} \cdot 1 = x^{12} \)- For \( k = 1 \): \( \binom{6}{1} (x^2)^5 (-y^2)^1 = 6 \cdot x^{10} \cdot (-y^2) = -6x^{10}y^2 \)- For \( k = 2 \): \( \binom{6}{2} (x^2)^4 (-y^2)^2 = 15 \cdot x^8 \cdot y^4 = 15x^8y^4 \)- For \( k = 3 \): \( \binom{6}{3} (x^2)^3 (-y^2)^3 = 20 \cdot x^6 \cdot (-y^6) = -20x^6y^6 \)- For \( k = 4 \): \( \binom{6}{4} (x^2)^2 (-y^2)^4 = 15 \cdot x^4 \cdot y^8 = 15x^4y^8 \)- For \( k = 5 \): \( \binom{6}{5} (x^2)^1 (-y^2)^5 = 6 \cdot x^2 \cdot (-y^{10}) = -6x^2y^{10} \)- For \( k = 6 \): \( \binom{6}{6} (x^2)^0 (-y^2)^6 = 1 \cdot 1 \cdot y^{12} = y^{12} \)

- Write the Final Expanded Expression

Combine all the expanded terms: \[ x^{12} - 6x^{10}y^2 + 15x^8y^4 - 20x^6y^6 + 15x^4y^8 - 6x^2y^{10} + y^{12} \]

Key concepts

Binomial Expansion

The Binomial Theorem allows us to expand expressions of the form \((a + b)^n\).This theorem is extremely useful when working with high powers of binomials, as it breaks them down into manageable terms.The general formula is given by: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \].
Each term in the expansion comes from taking different powers of \(a\) and \(b\),multiplied by the corresponding binomial coefficients.
For example, when applying the theorem to \((x^2 - y^2)^6\), the terms will include various powers of \(x^2\) and \(-y^2\).Understanding each part of this formula is key in performing binomial expansion.

Binomial Coefficients

Binomial coefficients are found in the expansion formula, represented as \( \binom{n}{k} \).These coefficients indicate how many ways terms can be selected and arranged. \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \].
\(n!\) represents the factorial of \(n\), calculated as the product of all positive integers up to \(n\).For instance, in our example of \((x^2 - y^2)^6\), \(n = 6\) and the binomial coefficients are calculated for each \(k\) from 0 to 6.
These calculations give us: \[ \binom{6}{0} = 1, \binom{6}{1} = 6, \binom{6}{2} = 15, \binom{6}{3} = 20, \binom{6}{4} = 15, \binom{6}{5} = 6, \binom{6}{6} = 1 \].
The coefficients help determine the weight of each term in the expanded expression.

Polynomial Expansion

A polynomial is an algebraic expression consisting of variables and coefficients, structured in terms like \(ax^n\).When we expand a binomial expression like \((x^2 - y^2)^6\) using the Binomial Theorem, we are generating a polynomial.
The expansion involves multiple terms of the form \(a_{i}x^{i}y^{j}\).Each term arises from the powers of \(a\) and \(b\), multiplied by their respective binomial coefficients.
In the exercise provided, substituting \(a = x^2\), \(b = -y^2\) and \(n = 6\) results in terms such as \(x^{12}\), \(-6x^{10}y^2\), and so forth.By writing each of these terms sequentially, we construct the expanded polynomial.

Algebraic Expressions

Algebraic expressions include variables, constants, and operations like addition, subtraction, multiplication, and division.Each term in an algebraic expression represents a part of the problem.
For instance, in \((x^2 - y^2)^6\), \((x^2 - y^2)\) is an algebraic expression that will be expanded as a polynomial.
When we apply the Binomial Theorem, we transform this algebraic expression into a sum of multiple terms.Each term will include variables \(x\) and \(y\) raised to various powers, and may also involve coefficients, as shown earlier.Handling algebraic expressions requires knowing how to manipulate variables and constants systematically, ensuring each step is logically consistent.
This process highlights the fundamental operations in algebra, underpinning much of mathematics.