Problem 63 - Algebra 2

Question

State the number of terms in each expansion and give the first two terms. $$ (2 a+b)^{5} $$

Step-by-Step Solution

Answer

The number of terms in the binomial expansion \((2a+b)^5\) is 6. The first two terms of the expansion are \(32a^5\) and \(80a^4b\).

Step 1: Identifying the Number of Terms

In a binomial expansion \((a+b)^n\), the number of terms is always \(n+1\). So, for the expansion \((2a+b)^5\), the number of terms is \(5+1=6\).

Step 2: Calculating the First Term

The first term of a binomial expansion is calculated by raising the first term in the binomial to the nth power. Hence, for the expansion \((2a+b)^5\), the first term is \((2a)^5 = 32a^5\).

Step 3: Calculating the Second Term

The second term of a binomial expansion is calculated using the formula \(\binom{n}{1} * a^{n-1} * b\). Hence, for the expansion \((2a+b)^5\), the second term is \(\binom{5}{1} * (2a)^4 * b = 5 * 16a^4 * b = 80a^4b\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a fundamental concept in algebra, particularly when dealing with expressions that involve powers of a binomial, like \((a+b)^n\). This theorem provides a formula to expand these expressions into a sum involving terms of the form \(a^k b^{n-k}\). It states that:
\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]This succinctly means:

• For each power \(k\), there's a coefficient \(\binom{n}{k}\), called a binomial coefficient, representing the number of ways to pick \(k\) elements from \(n\).
• The terms decrease in the power of \(a\) and increase in the power of \(b\) as you move through the expansion.

In any n-degree binomial expansion, the number of individual terms is always \(n+1\). Understanding this theorem gives a solid foundation for tackling similar polynomial expressions.

Algebra

Algebra acts as a language of symbols and rules, and it's incredibly powerful for solving equations and expressions. It provides the tools needed to understand and manipulate formulas, numbers, and equations.
In algebraic terms, expressions like \((2a + b)^5\) are explored to understand relationships and transformations. Specifically, algebra is useful in specifying how elements within an expression relate, whether multiplying a constant or understanding powers of variables.
Tools from algebra include:

• Understanding coefficients and like terms
• Applying rules of exponents
• Organizing complex expressions into manageable pieces

Algebra helps not just in dealing with polynomial expressions but is also essential for higher mathematics and real-world problems.

Polynomial Expansion

Polynomial Expansion involves expressing powers of polynomials as a series of terms. In an expansion like \((2a+b)^5\), understanding each term allows you to predict outcomes without multiplying everything directly.
This process transforms a potentially complicated expression into several individual components that
are simpler to handle. With the binomial theorem, you:

• Express every term with coefficients derived from the binomial formula.
• Recognize the symmetry of powers, where \(a\) starts with a high power decreasing by 1 each term, while \(b\) starts at zero, increasing by 1 each term.

Polynomial expansion is crucial for a variety of applications, from simplification tasks to seeking precise answers in engineering and physics problems.

Terms of a Polynomial

The terms of a polynomial refer to the separate parts of a polynomial expression that are added together. Each term is typically composed of a coefficient multiplied by variables raised to powers.
In expansions like \((2a+b)^5\), you get terms such as \(32a^5\) and \(80a^4b\), where each term results from operation on the original binomial expression.
Key points include:

• The number of terms is determined by the polynomial's degree, plus one.
• The sequential operations like coefficients, multiplication of terms, and appropriate power of variables are central to constructing each term.

Understanding terms helps in simplifying expressions, solving equations, and interpreting real-world contexts that rely on algebraic expressions.