Problem 75 - Algebra 2

Question

Expand each binomial. $$ (x+y)^{5} $$

Step-by-Step Solution

Answer

The expanded form of the binomial \( (x+y)^{5} \) is \[ x^{5} + 5x^{4}y + 10x^{3}y^{2} + 10x^{2}y^{3} + 5xy^{4} + y^{5} \]

Step 1: Using Binomial Theorem for Expansion

Our binomial will be expanded using the Binomial theorem as given by \( (a+b)^{n} = \sum_ {k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \). Here, \( a = x, b = y \) and \( n = 5 \).

Step 2: Calculate Each Term of the Expansion

The binomial theorem will help us find the coefficients of each term, the complete expansion of \( (x+y)^{5} \) turns out as follows each term in the sum : \\[ (x+y)^{5} = \binom{5}{0} x^{5} y^{0} + \binom{5}{1} x^{4} y^{1} + \binom{5}{2} x^{3} y^{2} + \binom{5}{3} x^{2} y^{3} + \binom{5}{4} x^{1} y^{4} + \binom{5}{5} x^{0} y^{5} \]

Step 3: Evaluating binomial coefficients and Simplification

The binomial coefficients need to be evaluated using the binomial coefficient formula \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]. After simplification of these expressions the answer is: \[ x^{5} + 5x^{4}y + 10x^{3}y^{2} + 10x^{2}y^{3} + 5xy^{4} + y^{5} \]

Key Concepts

Binomial Expansion

The concept of binomial expansion refers to expressing a binomial expression, like \((x + y)^5\), in its expanded polynomial form. The Binomial Theorem is pivotal in this process. It provides a formula to expand powers of binomials into polynomial terms. This formula is:\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]where:

• n is the power to which the binomial is raised
• \(\binom{n}{k}\) are the binomial coefficients
• a and b are the terms in the binomial

In the binomial expansion \( (x+y)^5 \), each term is defined by incrementally increasing the power of \(y\) and decreasing the power of \(x\), while coefficients for each term are calculated based on combinatorial methods such as Pascal's Triangle or the binomial coefficient formula.

Combinatorics

Combinatorics plays a crucial role in binomial expansion through the calculation of binomial coefficients. These coefficients essentially count the number of ways \(k\) terms can be chosen from \(n\) terms. The mathematical formula for this is given by:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]where:

• \(n!\) (n factorial) is the product of all positive integers up to \(n\)
• \(k!\) is the factorial of \(k\)
• (n-k)! is the factorial of \(n-k\)

For example, to compute \(\binom{5}{2}\) in the expansion of \((x+y)^5\), you would use:\[ \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \]Combinations help us determine the right coefficients for each term in the binomial expansion, ensuring the expansion adheres to the binomial theorem structure.

Polynomial Expansion

Polynomial expansion is the process of expressing a binomial raised to a power as a sum of monomial terms. Each term results from the multiplication of the base expression with coefficients obtained from the binomial theorem.In the case of \((x+y)^5\), the expansion results in:\[ x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5 \]Each term's degree is equal to the original power, in this case, 5. The exponents of \(x\) and \(y\) decrease and increase respectively, from term to term.
Polynomial expansions allow us to see the full distribution of terms and their coefficients, simplifying calculations and understanding the impact of raising binomials to powers.