Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-10,\) write the expansion of each binomial. $$ (a+3)^{4} $$

Step-by-Step Solution

Answer

The expansion of \((a+3)^4\) is \(a^4 + 12a^3 + 54a^2 + 108a + 81\).

Step 1: Identify the Binomial Form

The given expression is a binomial raised to the fourth power, \[(a+3)^4\]. This fits the form \[(x + y)^n\], where \( x = a \), \( y = 3 \), and \( n = 4 \).

Step 2: Apply the Binomial Theorem

The Binomial Theorem states that \[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} \, x^{n-k} \, y^k\].Apply this to our expression with \(a\) and \(3\):\[(a+3)^4 = \sum_{k=0}^{4} \binom{4}{k} \, a^{4-k} \, 3^k\].

Step 3: Calculate Each Term of the Expansion

Calculate each term using the Binomial Coefficient \[\binom{4}{k}\] and the powers of \(a\) and 3: 1. \(k=0\): \(\binom{4}{0} \, a^{4} \, 3^0 = 1\cdot a^4\cdot 1 = a^4\)2. \(k=1\): \(\binom{4}{1} \, a^{3} \, 3^1 = 4\cdot a^3\cdot 3 = 12a^3\)3. \(k=2\): \(\binom{4}{2} \, a^{2} \, 3^2 = 6\cdot a^2\cdot 9 = 54a^2\)4. \(k=3\): \(\binom{4}{3} \, a^{1} \, 3^3 = 4\cdot a\cdot 27 = 108a\)5. \(k=4\): \(\binom{4}{4} \, a^{0} \, 3^4 = 1\cdot 1\cdot 81 = 81\)

Step 4: Combine the Calculated Terms

Add all the calculated terms together to form the complete expansion:\[(a+3)^4 = a^4 + 12a^3 + 54a^2 + 108a + 81\]

Key Concepts

Binomial Theorem

The Binomial Theorem is an essential tool in algebra for expanding expressions that are raised to a power. It provides a formula that allows us to calculate the expansion of any binomial expression of the form \((x + y)^n\) in a systematic manner. The theorem states that \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} \, x^{n-k} \, y^k\), where:

• \(n\) is the power to which the binomial is raised,
• \(x\) and \(y\) are any numbers, variables, or expressions,
• \(\binom{n}{k}\) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\).

By applying these simple rules, you can expand any expression of this form. For example, in \((a+3)^4\), the Binomial Theorem helps identify and calculate each term's coefficients and the corresponding powers of \(a\) and \(3\) without excessive computation.

Binomial Coefficients

Binomial coefficients are crucial in the Binomial Theorem, where they show up in the expression \(\binom{n}{k}\). These coefficients indicate the number of ways to pick \(k\) elements from a total of \(n\) without regard to order. They are calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Understanding binomial coefficients is essential because they determine how much each term in a binomial expansion is "weighted." In our example, \(\binom{4}{2}\) in the expansion of \((a+3)^4\) contributes to the term \(54a^2\). To calculate it:

• Calculate the factorials: \(4! = 24\), \(2! = 2\), and \((4-2)! = 2! = 2\).
• Apply the formula: \(\binom{4}{2} = \frac{24}{2 \times 2} = 6\).

These coefficients determine the term's contribution in terms of magnitude relative to other terms in a binomial expansion.

Polynomial Expansion

Polynomial expansion refers to breaking down complex binomial expressions into simpler polynomial forms using rules such as the Binomial Theorem. It involves transforming a single binomial power expression into a series of terms added together, each with coefficients and designated powers. Polynomial expansion is widely used in algebra to simplify problems involving squares, cubes, or higher powers of binomials.For example, the expansion of \((a+3)^4\) results in a polynomial \(a^4 + 12a^3 + 54a^2 + 108a + 81\), where each term is carefully calculated using binomial coefficients. A polynomial here is the sum of terms of the form \(ca^m\), where \(c\) is a coefficient, and each \(m\) represents the power resulting from expanding the binomial.Polynomial expansions help visualize how each component of a binomial contributes to the overall expression, which is beneficial in understanding algebraic constructions and solving equations.

Power of a Binomial

The term "power of a binomial" refers to raising a binomial expression to a specified exponent in mathematics. It represents how many times the binomial is multiplied by itself. Each power results in a specific polynomial with a varying number of terms and complexity depending on the power itself.When you have a binomial raised to a high power, like \((a+3)^4\), manually multiplying it repeatedly can be tedious and prone to error. Instead, techniques like the Binomial Theorem provide useful shortcuts to facilitate this expansion into a neat polynomial.In the example \((a+3)^4\), each term is formed by applying the binomial coefficients and multiplying different powers of \(a\) and \(3\). This method ensures that you efficiently capture all possible products between the terms of \(a\) and \(3\) while respecting the degree given by the power, making binomial expansion manageable and systematic.