Problem 41 - Algebra 2

Question

Expand each binomial. $$ (x-3 y)^{6} $$

Step-by-Step Solution

Answer

The expanded form of \((x - 3y)^6\) is \(x^{6} - 18x^{5} y + 135x^{4} y^{2} - 540x^{3} y^{3} + 1215x^{2} y^{4} - 1458x y^{5} + 729 y^{6}\)

Step 1: Binomial Theorem

The binomial theorem states that for any number \(a\), \(b\), and integer \(n\): \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]In this case, \(a\) is \(x\), \(b\) is \(-3y\), and \(n\) is \(6\).

Step 2: Calculating the Binomial Coefficients

The binomial coefficient can be represented by \( \binom{n}{k} \): When \(k=0\), \(\binom{6}{0} = 1\),when \(k=1\), \(\binom{6}{1} = 6\),when \(k=2\), \(\binom{6}{2} = 15\),when \(k=3\), \(\binom{6}{3} = 20\),when \(k=4\), \(\binom{6}{4} = 15\),when \(k=5\), \(\binom{6}{5} = 6\),when \(k=6\), \(\binom{6}{6} = 1\).

Step 3: Applying Coefficients and Expanding

From the binomial theorem, the expanded form of \((x-3y)^6\) can be calculated as - \[(x-3y)^6 = \binom{6}{0} x^{6} (-3y)^{0} + \binom{6}{1} x^{5} (-3y)^{1} + \binom{6}{2} x^{4} (-3y)^{2} + \binom{6}{3} x^{3} (-3y)^{3} + \binom{6}{4} x^{2} (-3y)^{4} + \binom{6}{5} x^{1} (-3y)^{5} + \binom{6}{6} x^{0} (-3y)^{6}\]which evaluates into \[x^{6} - 6x^{5} (3y) + 15x^{4} (9y^2) - 20x^{3} (27y^3) + 15x^{2} (81y^4) - 6x (243y^5) + (729y^6)\]

Step 4: Simplifying

This can be further simplified to -\[x^{6} - 18x^{5} y + 135x^{4} y^{2} - 540x^{3} y^{3} + 1215x^{2} y^{4} - 1458x y^{5} + 729 y^{6}\]

Key Concepts

Binomial Expansion

Binomial expansion is a process used to expand expressions that are raised to a power, such as \((a+b)^n\). The Binomial Theorem provides a formula to do this efficiently.
It reveals how each term in the expanded polynomial can be found.

• The theorem states that each term is constructed by multiplying the binomial coefficient with the powers of the two terms in the binomial.
• For example, in the expression \((x-3y)^6\), each term of the expansion is influenced by the powers of both \(x\) and \(-3y\).

The Binomial Theorem is especially helpful when \(n\) is a relatively small positive integer, as it allows for a straightforward method
to expand to get the full polynomial form.

Binomial Coefficient

A binomial coefficient, denoted as \(\binom{n}{k}\), is a critical component in binomial expansion. It determines the weight of each term in the expansion.
The coefficient \(\binom{n}{k}\) essentially counts the number of ways to choose \(k\) elements from a set of \(n\) elements. These coefficients can be found using factorials, where:

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

For the problem \((x-3y)^6\), the coefficients are found for each \(k\) from 0 to 6.
The pattern observed in Pascal's Triangle can also help in determining these coefficients without manual computation.

Polynomial Expansion

Polynomial expansion involves expressing a binomial or a higher-order polynomial
as a sum of simpler terms. In the example of \((x-3y)^6\), this expansion includes several terms which are each a combination of a binomial coefficient, powers of \(x\), and powers of \(-3y\).

• The resulting polynomial includes terms like \(x^6\), \(-18x^5y\), and so on, down to \(729y^6\).
• Each term involves multiplying the binomial coefficient by appropriate powers of \(x\) and \(y\), and by the magnitude of \((-3)^k\).

Simplifying this expansion involves multiplying these components and combining like terms,
resulting in a polynomial that accurately represents the expanded form of the original binomial expression.