Problem 39 - Algebra 2

Question

fourth term of \((c+d)^{8}\)

Step-by-Step Solution

Answer

The fourth term is 56c^5d^3.

Step 1: Understand the Binomial Expansion

The expression (c+d)^{8} is a binomial expression that can be expanded using the binomial theorem. According to the binomial theorem, the expansion of (a+b)^n is given by: \(\sum_{k=0}^{n} C(n,k) a^{n-k} b^k\) where C(n,k) is the binomial coefficient \(\frac{n!}{k!(n-k)!}\).

Step 2: Identify the Fourth Term

The general term of a binomial expansion \(T_{k+1}\) is given by: \(T_{k+1} = C(n,k) a^{n-k} b^k\). For the fourth term, kequals 3. So, identify the components: \(a=c\), \(b=d\), and \(n=8\).

Step 3: Calculate the Binomial Coefficient for the Fourth Term

Calculate the binomial coefficient C(8,3) as follows: \(C(8,3) = \frac{8!}{3!(8-3)!}\) \(\frac{8!}{3!5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56\).

Step 4: Formulate the Fourth Term Using the Binomial Coefficient

Using the formula for the general term with the values identified: \(T_4 = C(8,3) \, c^{8-3} \, d^3\) \(= 56 \, c^5 \, d^3\).

Key Concepts

Binomial Expansion

When you see a binomial expression, like every long paragraph. (c+d)^8, you can expand it to include several terms using the binomial theorem. This means taking a power, in this case, 8, and breaking it down into separate terms. It helps you find expressions like (e+f)^5 without doing a complete manual multiplication.

The binomial theorem gives us a formula to expand a binomial expression. This formula expresses (a+b)^n as a sum:\[\sum_{k=0}^{n} C(n,k) a^{n-k} b^k\]where a, b are the terms of your binomial and n is the power.

This process is particularly beneficial:

• For expanding expressions in a detailed manner.
• To find specific term values.
• In problems involving probability, sequences, and series in algebra.

Binomial Coefficient

In the binomial expansion, the binomial coefficient plays a vital role. It tells you the weight of each term in the expanded form. In simpler terms, it determines how much each term appears in the expansion.

The binomial coefficient is represented by:\[C(n,k) = \frac{n!}{k!(n-k)!}\] where n is the exponent, k is the term number, and ! indicates a factorial.

Consider our example:

• For the expression \((c+d)^8\), calculating the fourth term requires the coefficient \(C(8,3)\).
• We determine it by substituting n and k into the formula: \(C(8,3) = \frac{8!}{3! (8-3)!}\).
• This evaluates to 56, which scales the term \(c^5d^3\) in the expansion.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operations. In algebra, these expressions help represent relationships and solve equations.

Consider the binomial expression \((c+d)^8\), which is a specific type of algebraic expression involving two terms, c and d. By studying how to expand it, you simplify complex expressions and explore their structure.

In our example, becoming comfortable with algebraic expressions involves:

• Understanding how terms like \(c^5d^3\) are formed when expressions are expanded.
• Recognizing how each component, including coefficients, variables, and powers, contributes to forming expanded expressions.
• Utilizing expanded expressions to solve larger algebraic problems or explore sequences and combinations.

Algebraic expressions form the basis for many advanced mathematical concepts, making their understanding essential.