Problem 36 - Algebra for College Students

Question

Expand each binomial and simplify.\((x+2 y)^{5}\)

Step-by-Step Solution

Answer

Expanded form: \(x^5 + 10x^4y + 40x^3y^2 + 80x^2y^3 + 80xy^4 + 32y^5\).

Step 1: Identify the Binomial Theorem

The binomial theorem is a useful formula for expanding binomials of the form \((a + b)^n\). It states: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is a binomial coefficient. In this problem, \(a = x\), \(b = 2y\), and \(n = 5\).

Step 2: Compute Binomial Coefficients

To expand \((x + 2y)^5\), we need the coefficients \(\binom{5}{0}, \binom{5}{1}, \binom{5}{2}, \binom{5}{3}, \binom{5}{4}, \) and \(\binom{5}{5}\). These coefficients are calculated as follows:\(\binom{5}{0} = 1\)\(\binom{5}{1} = 5\)\(\binom{5}{2} = 10\)\(\binom{5}{3} = 10\)\(\binom{5}{4} = 5\)\(\binom{5}{5} = 1\).

Step 3: Apply the Binomial Formula

Substitute the values into the binomial expansion formula:\((x+2y)^5 = \sum_{k=0}^{5} \binom{5}{k} x^{5-k} (2y)^k\).

Step 4: Calculate Each Term

Compute each term by substituting \(x\) and \(2y\) into the equation:- For \(k = 0\): \(\binom{5}{0} x^5 (2y)^0 = 1 \cdot x^5 \cdot 1 = x^5\).- For \(k = 1\): \(\binom{5}{1} x^4 (2y)^1 = 5 \cdot x^4 \cdot 2y = 10x^4y\).- For \(k = 2\): \(\binom{5}{2} x^3 (2y)^2 = 10 \cdot x^3 \cdot 4y^2 = 40x^3y^2\).- For \(k = 3\): \(\binom{5}{3} x^2 (2y)^3 = 10 \cdot x^2 \cdot 8y^3 = 80x^2y^3\).- For \(k = 4\): \(\binom{5}{4} x^1 (2y)^4 = 5 \cdot x^1 \cdot 16y^4 = 80xy^4\).- For \(k = 5\): \(\binom{5}{5} x^0 (2y)^5 = 1 \cdot 1 \cdot 32y^5 = 32y^5\).

Step 5: Combine All Terms

Combine all calculated terms to find the expanded and simplified form of \((x + 2y)^5\):\(x^5 + 10x^4y + 40x^3y^2 + 80x^2y^3 + 80xy^4 + 32y^5\).

Key Concepts

Binomial Expansion

The binomial expansion is a way to express the power of a binomial, like \((a + b)^n\), as a sum of terms. Each term is made up of the binomial coefficients, a power of \(a\), and a power of \(b\). These terms are sourced from the Binomial Theorem. This theorem is very useful in simplifying the process of raising binomials to a large power.

To use the theorem, first identify the elements:

• \(a\) and \(b\) are the two terms in the binomial.
• \(n\) is the power the binomial is raised to.
• The formula for expansion is \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\).

This formula shows that each term in the expansion requires calculating binomial coefficients and applying them to the powers of \(a\) and \(b\). The sum of all the terms gives the expanded polynomial.

Exponentiation

Exponentiation is the mathematical operation of raising one number, known as the base, to the power of another number. In the context of binomial expansion, the exponents apply to the terms in the binomial expression \((a + b)^n\).

Each term in the expansion has two components that involve exponents:

• The first term: \(a^{n-k}\)
• The second term: \(b^k\)

- The exponent \(n-k\) is applied to the first term, \(a\),ensuring it is raised in successive descending powers. - The exponent \(k\) is applied to the second term, \(b\), increasing its power in each subsequent term. Together, these exponents control how the original binomial is transformed into its expanded series of terms.

Binomial Coefficient

The binomial coefficient is crucial in binomial expansions. They determine the multipliers for each term in the expansion of a binomial. The binomial coefficient is denoted as \(\binom{n}{k}\), which reads as "n choose k."

This coefficient calculates how many ways you can pick \(k\) items from \(n\) items without regard to order. Mathematically, it is expressed as: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Here

• \(!\) is the factorial operation, which multiplies a series of descending natural numbers.
• \(n!\) is the factorial of \(n\).
• \(k!\) and \((n-k)!\) are the factorials of \(k\) and \(n-k\) respectively.

These coefficients essentially weigh how much each term contributes to the final expression.

Polynomial Expansion

Polynomial expansion is a method where a binomial is raised to a power and expanded into a sum of individual monomial terms. This is what occurs when the Binomial Theorem is applied to expressions like \((x + 2y)^5\).

The result of applying the Binomial Theorem is a polynomial, which consists of several terms including:

• The first term: the highest power of \(x\).
• The middle terms: mixed powers of \(x\) and \(y\).
• The last term: the highest power of \(y\).

Each of these terms is the outcome of applying binomial coefficients and exponentiation to the original binomial. This method is powerful and efficient for large exponents, making it rather straightforward to expand complex binomials into simpler polynomial forms.