Problem 38 - Algebra 2

Question

PREREQUISITE SKILL Find the indicated term of each expression. third term of \((a+b)^{7}\)

Step-by-Step Solution

Answer

The third term is \(21a^5b^2\).

Step 1: Understand Binomial Expansion

The expression \((a+b)^n\) is expanded using the binomial theorem. The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient.

Step 2: Identify the Formula for the Binomial Coefficient

The binomial coefficient, \(\binom{n}{k}\), is calculated as \(\frac{n!}{k!(n-k)!}\). This represents the number of ways to choose \(k\) elements from \(n\) elements.

Step 3: Determine the Required Term's Position

For the third term of the expansion, set \(k = 2\). Binomial expansions are indexed starting from 0, so the third term corresponds to \(k=2\).

Step 4: Calculate the Binomial Coefficient for the Third Term

Substitute \(n = 7\) and \(k = 2\) into the binomial coefficient formula: \(\binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21\). This is the coefficient for the third term.

Step 5: Find the Powers of Terms for the Third Term

The expression has the form \(\binom{7}{2} a^{n-k} b^k\). Here, \(a^{7-2} = a^5\) and \(b^2\).

Step 6: Write Down the Third Term

Combine the binomial coefficient and the powers of \(a\) and \(b\): The third term is \(21a^5b^2\).

Key Concepts

Binomial Expansion

When we expand expressions of the form \(a + b\)^n\, we use the **binomial expansion**. This concept helps to systematically expand binomials without having to manually multiply them. The binomial theorem provides a formula that tells us how to expand any power of a binomial expression easily. The expansion is expressed as:\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Here's how you do it step by step:- Identify the desired power of the binomial expression.- Use the formula to find each term of the expansion by varying `k` from 0 up to `n`.- Each term in the expansion is composed of a binomial coefficient, a power of `a`, and a power of `b`.This method ensures a systematic and efficient way of expanding binomials, making it especially helpful for larger powers.

Binomial Coefficient

The **binomial coefficient** is a crucial part in binomial expansions. It is denoted as \ \binom{n}{k} \, which reads as "n choose k". It represents the number of ways to choose `k` items from a total of `n` items, disregarding order.The formula for calculating it is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]**What Does This Mean?**- **n! (n factorial):** This is the product of all positive integers upto `n`.- **k! (k factorial):** The product of all positive integers up to `k`.- **(n-k)!** is similarly calculated for `n-k`.By applying this formula, you can precisely determine the coefficient for each term of a binomial expansion.

Algebraic Expressions

**Algebraic expressions** consist of variables and constants combined using operations like addition, subtraction, multiplication, and division. In the context of binomial expansion, the expressions \ (a+b) \ are manipulated using algebraic techniques.### Components of Algebraic Expressions:- **Terms:** These are the separate parts of an expression. In the expression \(a + b\)^2\, the simplified terms are \ a^2\, \ 2ab\, and \ b^2\.- **Factors:** Terms themselves can consist of factors, such as \ a^2 \ being \ a \ multiplied by \ a\.- **Coefficients:** These are numbers in front of the variables that multiply with them, such as 21 in \ 21a^5b^2\.Mastering algebraic expressions is pivotal since they form the foundation for more advanced concepts in mathematics, including binomial expansions.