Problem 27 - Algebra and Trigonometry

Question

Expand each expression using the Binomial Theorem. $$ (a x+b y)^{5} $$

Step-by-Step Solution

Answer

The expanded form is \(a^5 x^5 + 5a^4 x^4 b y + 10a^3 x^3 b^2 y^2 + 10a^2 x^2 b^3 y^3 + 5a x b^4 y^4 + b^5 y^5\).

Step 1: Identify the components

In the expression \((ax + by)^5\), identify the coefficients and variables: \(a\), \(b\), \(x\), and \(y\). The binomial theorem states that \[(ax + by)^n = \sum_{k=0}^{n} \binom{n}{k} (ax)^{n-k} (by)^k\] where \(n = 5\) in this case.

Step 2: Apply the Binomial Theorem

Use the binomial theorem formula with \(n = 5\). Expand the sum where \(k\) ranges from 0 to 5:\[(ax + by)^5 = \sum_{k=0}^{5} \binom{5}{k} (ax)^{5-k} (by)^k\].

Step 3: Calculate each term

Calculate the binomial coefficient and each term for \(k = 0, 1, 2, 3, 4, 5\).\[(ax + by)^5 = \binom{5}{0} (ax)^5 (by)^0 + \binom{5}{1} (ax)^4 (by)^1 + \binom{5}{2} (ax)^3 (by)^2 + \binom{5}{3} (ax)^2 (by)^3 + \binom{5}{4} (ax)^1 (by)^4 + \binom{5}{5} (ax)^0 (by)^5\]

Step 4: Simplify each term

Simplify each term using binomial coefficients and powers:\[\binom{5}{0} (ax)^5 (by)^0 = 1 \cdot (ax)^5 \cdot 1 = a^5 x^5\]\[\binom{5}{1} (ax)^4 (by)^1 = 5 \cdot (ax)^4 \cdot by = 5a^4 x^4 b y\]\[\binom{5}{2} (ax)^3 (by)^2 = 10 \cdot (ax)^3 \cdot (by)^2 = 10a^3 x^3 b^2 y^2\]\[\binom{5}{3} (ax)^2 (by)^3 = 10 \cdot (ax)^2 \cdot (by)^3 = 10a^2 x^2 b^3 y^3\]\[\binom{5}{4} (ax)^1 (by)^4 = 5 \cdot (ax) \cdot (by)^4 = 5a x b^4 y^4\]\[\binom{5}{5} (ax)^0 (by)^5 = 1 \cdot 1 \cdot (by)^5 = b^5 y^5\]

Step 5: Combine the terms

Combine all simplified terms to get the final expansion:\[(ax + by)^5 = a^5 x^5 + 5a^4 x^4 b y + 10a^3 x^3 b^2 y^2 + 10a^2 x^2 b^3 y^3 + 5a x b^4 y^4 + b^5 y^5\]

Key Concepts

term expansion

The concept of term expansion involves expressing a polynomial or binomial raised to a power as a sum of terms. Each term in this expansion is derived from multiplying the terms of the original expression, each raised to different powers.

In the given problem, we start with the binomial \((ax + by)^5\). To expand this using the Binomial Theorem, we break it down into individual terms, each representing a different combination of the elements \(ax\) and \(by\), as raised to certain powers.

The term expansion unfolds into terms of the form \( \binom{n}{k} (ax)^{n-k} (by)^k \), where the exponent, powers, and coefficients all vary according to the binomial rules. This creates a polynomial expression in which the exponents of x and y add up to the power to which the binomial was raised.

By expanding the given binomial expression, we get a series of progressively simpler terms that, when added together, give us the full expanded form.

binomial coefficients

Binomial coefficients are key components in the Binomial Theorem, represented as \( \binom{n}{k} \). These coefficients essentially tell us how many ways we can pick \(k\) elements from \(n\) elements without regard to order.

For the expansion \((ax + by)^5\), we derive binomial coefficients for each value of \(k\) from 0 to 5. These coefficients are calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \(n\) is 5 in our specific example, and \(k\) varies from 0 to 5.

As demonstrated in the solution, the coefficients \( \binom{5}{k} \) become values like 1, 5, 10, and so forth, determining the weights of the corresponding terms in the expansion process. Each coefficient modifies how much of the \( (ax)^{n-k}\times(by)^k \) terms contribute to the final expanded polynomial.

polynomial expressions

Polynomial expressions are mathematical phrases involving sums of powers in one or more variables. They can come from expanding binomials like in our example with \((ax + by)^5\).

When expanded, this binomial polynomial expression becomes \(a^5 x^5 + 5a^4 x^4 b y + 10a^3 x^3 b^2 y^2 + 10a^2 x^2 b^3 y^3 + 5a x b^4 y^4 + b^5 y^5 \). Each term in this expanded polynomial is a product of constants (derived from the binomial coefficients) and the variables \(x\) and \(y\), each raised to certain powers.

In polynomial terms, the exponents of \(x\) and \(y\) add up to the original power of the binomial, which is 5 in this case. Remembering these rules helps in systematically expanding any polynomial expression using the Binomial Theorem. This understanding is essential in algebra and provides a foundation for further studies in calculus and higher mathematics.