Problem 28 - Algebra and Trigonometry

Question

Expand each expression using the Binomial Theorem. $$ (a x-b y)^{4} $$

Step-by-Step Solution

Answer

(ax - by)^4 = a^4 x^4 - 4a^3 b x^3 y + 6a^2 b^2 x^2 y^2 - 4a b^3 x y^3 + b^4 y^4.

Step 1: Write down the Binomial Theorem

The Binomial Theorem states that \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]. In this exercise, the expression is \( (ax - by)^4 \), so the theorem can be applied by considering \( a = ax \) and \( b = -by \).

Step 2: Identify the components

Identify \( a = ax \) and \( b = -by \), and \( n = 4 \). Plug these values into the formula: \[ (ax - by)^4 = \sum_{k=0}^{4} \binom{4}{k} (ax)^{4-k} (-by)^k \].

Step 3: Calculate each term in the summation

Calculate each term using the binomial coefficients: \[ \begin{aligned} & = \binom{4}{0}(ax)^4(-by)^0 + \binom{4}{1}(ax)^3(-by)^1 + \binom{4}{2}(ax)^2(-by)^2 + \binom{4}{3}(ax)^1(-by)^3 + \binom{4}{4}(ax)^0(-by)^4 \ & = 1 (ax)^4 + 4 (ax)^3 (-by) + 6 (ax)^2 (-by)^2 + 4 (ax) (-by)^3 + 1 (-by)^4 \ & = a^4 x^4 - 4a^3 x^3 b y + 6a^2 x^2 b^2 y^2 - 4a x b^3 y^3 + b^4 y^4 \end{aligned} \]

Step 4: Simplify the expression

Combine the like terms and ensure signs are correct: \[ (ax - by)^4 = a^4 x^4 - 4a^3 b x^3 y + 6a^2 b^2 x^2 y^2 - 4a b^3 x y^3 + b^4 y^4 \]

Key Concepts

binomial expansion

The binomial expansion is a way to expand expressions that are raised to a power, like \( (ax - by)^4 \). It uses the Binomial Theorem to break the expression into a sum of terms.

The Binomial Theorem states: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]. This formula gives each term of the expansion by combining powers of \(a\) and \(b\) with coefficients that are binomial coefficients.

To use the Binomial Theorem, you:

• Identify the variables \(a\), \(b\), and \(n\).


• Use the theorem to write the expanded form.


• Calculate each term.

It's helpful for simplifying expressions and for solving combinatorics problems.

binomial coefficients

Binomial coefficients are the numbers that appear in the binomial expansion. They are represented as \( \binom{n}{k} \) and are sometimes called 'combinations' or 'n choose k'.

These coefficients count the number of ways to choose \(k\) items from \(n\) items, and can be found using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] For example, if you want to find \( \binom{4}{2} \):

• Plug in the values: \( n = 4\) and \( k = 2 \).


• Calculate \(4!\), \(2!\), and \( (4-2)! \).


• Simplify: \( \binom{4}{2} = \frac{4!}{2!2!} = 6 \).

Using these coefficients in the Binomial Theorem lets you accurately expand expressions like \( (ax - by)^4 \).

expressions

In algebra, an expression is a combination of variables, constants, and operators like +, -, etc. For example, \( (ax - by)^4 \) is an expression that involves variables \(a, b, x,\) and \( y \), and a constant exponent 4.

When asked to expand an expression using the Binomial Theorem, follow these steps:

• Identify each part of the expression and any constants or variables present. For \( (ax - by)^4 \), \( a, b \, x \, y \, \text{and} 4 \).


• Rewrite the expression according to the Binomial Theorem.


• Calculate each term separately and combine them to form the expanded version. This results in terms like \( a^4 x^4, -4a^3 b x^3 y, \text{and so on} \).

Understanding expressions is key for algebra and calculus, as they form the basis for equations and functions.