Problem 23 - College Algebra

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (c+2)^{5} $$

Step-by-Step Solution

Answer

The expression \((c+2)^{5}\) simplifies to \(c^5 + 10c^4 + 40c^3 + 80c^2 + 80c + 32\).

Step 1: Understanding the Binomial Theorem

The Binomial Theorem helps to expand expressions of the form \((a+b)^n\), where 'n' is a natural number. In a more detailed form, \((a+b)^n = a^n + nC1 \cdot a^{n-1}b + nC2 \cdot a^{n-2}b^2 + .... + nCn \cdot b^n\). Here, 'nCr' denotes binomial coefficients, which can be found in Pascal's triangle or calculated using the formula n!/(r!(n-r)!).

Step 2: Identify a, b, and n

In the given expression \((c+2)^{5}\), 'c' is in place of 'a', '2' is in place of 'b', and '5' is in place of 'n'.

Step 3: Expand using the Binomial Theorem

Let's expand the expression now using the Binomial Theorem. So it becomes \(c^5 + 5C1 \cdot c^4 \cdot 2 + 5C2 \cdot c^3 \cdot 2^2 + 5C3 \cdot c^2 \cdot 2^3 + 5C4 \cdot c^1 \cdot 2^4 + 5C5 \cdot 2^5\).

Step 4: Substitute the binomial coefficients

Replace binomial coefficients 5C1, 5C2, .. with their respective values; i.e., 5, 10, 10, 5, 1. Now we get \(c^5 + 5 \cdot c^4 \cdot 2 + 10 \cdot c^3 \cdot 2^2 + 10 \cdot c^2 \cdot 2^3 + 5 \cdot c \cdot 2^4 + 2^5\).

Step 5: Simplify the Expression

Unroll multiplications to simplify the expression which gives us \(c^5 + 10c^4 + 40c^3 + 80c^2 + 80c + 32\).

Key Concepts

Binomial Expansion

The binomial expansion is a powerful mathematical tool, allowing us to expand expressions of the form \((a+b)^n\) into a series of terms. The expansion is done using the Binomial Theorem, which provides a systematic way to distribute the powers and binomial coefficients across the terms. Each term in the expansion consists of a power of 'a' and a power of 'b', with their exponents always adding up to 'n'. For example, with \((c+2)^5\), the expansion results in six terms: \(c^5, c^4, c^3, c^2, c,\) and a constant (from \(2^5\)).
The beauty of the binomial expansion lies in how it simplifies complex polynomial expressions. It finds applications in various fields such as probability, algebra, and even calculus.
Understanding the expansion process not only improves mathematical skills but also provides insight into how polynomial expressions can be manipulated.

Binomial Coefficients

Binomial coefficients are the numerical factors that accompany each term in a binomial expansion. Denoted traditionally as \(nCr\), these coefficients signify the number of ways to choose 'r' elements from a set of 'n' elements without regard to order. In simpler terms, they are the numbers that multiply the terms as you expand a binomial expression.
For example, when expanding \((c+2)^5\), the coefficients are derived from 5 choose 0, 5 choose 1, etc. The sequence is: 1, 5, 10, 10, 5, 1. These values can be easily found using Pascal's triangle or computed using the formula \(\frac{n!}{r!(n-r)!}\).
Binomial coefficients are pivotal as they determine the weight of each term in the expansion, ensuring every combination of powers is accurately represented.

Pascal's Triangle

Pascal's triangle is a simple yet effective triangular array of binomial coefficients. Each row corresponds to the coefficients found in the expansion of a binomial raised to a power that matches the row number. Starting with row 0 at the top, each number is the sum of the two directly above it in the previous row.
In the context of our example, for \((c+2)^5\), the coefficients (1, 5, 10, 10, 5, 1) can be directly extracted from the sixth row of Pascal's triangle. This is because the row index starts at 0, aligning with the row produced for a power of five in the binomial expansion.
Using Pascal's triangle simplifies finding coefficients without having to perform the complete factorial computation each time, speeding up the process and minimizing human error.

Factorials

Factorials, denoted as \(n!\), represent the product of all whole numbers from 1 to 'n'. They play a crucial role in the calculation of binomial coefficients. Factorials help in determining the number of possible combinations of elements, which is why they are pivotal in combinatorics and associated with permutations and combinations.
For a number 'n', the factorial is calculated as \(n \times (n-1) \times (n-2) \times ... \times 1\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
In the context of our binomial coefficient, \(nCr\) is calculated using \(\frac{n!}{r!(n-r)!}\). This formula illustrates how factorials help to find the specific number that serves as a coefficient in a binomial expansion. Mastery of this concept is key for advanced algebra and calculus.