Problem 41 - Beginning and Intermediate Algebra

Question

Use the binomial theorem to expand each expression. $$(3 m+2)^{4}$$

Step-by-Step Solution

Answer

The expanded expression of \((3m+2)^4\) using the Binomial Theorem is \(81m^4 + 72m^3 + 108m^2 + 64m + 16\).

Step 1: Identify the Binomial Coefficient

The binomial coefficient is given by the formula: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\] In our case, n=4 since the given expression is raised to the power of 4.

Step 2: Observe Pascal's Triangle

The 4th row of Pascal's Triangle gives us the coefficients for each term in the expansion of the binomial expression. The 4th row of Pascal's Triangle is: \[1\enspace 4\enspace 6\enspace 4\enspace 1\]

Step 3: Apply the Expansion

Now we can expand the expression \((3m+2)^4\) with the help of the binomial coefficients as follows: \[\begin{aligned} (3m+2)^4 &= \binom{4}{0}(3m)^4(2)^0 + \binom{4}{1}(3m)^3(2)^1 + \binom{4}{2}(3m)^2(2)^2 \\ &\phantom{=}+ \binom{4}{3}(3m)^1(2)^3 + \binom{4}{4}(3m)^0(2)^4 \end{aligned}\]

Step 4: Compute Binomial Coefficients

Now, we will compute the binomial coefficient for each term using the formula and the coefficients from Pascal's Triangle: \[\begin{aligned} &\binom{4}{0} = \frac{4!}{0!(4-0)!} = 1, \quad &\binom{4}{1} = \frac{4!}{1!(4-1)!} = 4, \\ &\binom{4}{2} = \frac{4!}{2!(4-2)!} = 6, \quad &\binom{4}{3} = \frac{4!}{3!(4-3)!} = 4, \\ &\binom{4}{4} = \frac{4!}{4!(4-4)!} = 1. \end{aligned}\]

Step 5: Simplify the Expansion

Substitute the computed binomial coefficients into the expanded expression and simplify: \[\begin{aligned} (3m+2)^4 &= 1(3m)^4(2)^0 + 4(3m)^3(2)^1 + 6(3m)^2(2)^2 + 4(3m)^1(2)^3 + 1(3m)^0(2)^4 \\ &= (3m)^4 + 4(3m)^3(2) + 6(3m)^2(2)^2 + 4(3m)(2)^3 + (2)^4 \\ &= 81m^4 + 72m^3 + 108m^2 + 64m + 16 \end{aligned}\] So, the expanded expression of $$(3m+2)^4$$ is $$81m^4 + 72 m^3 + 108m^2 + 64m + 16$$.

Key Concepts

Binomial Coefficients

The concept of binomial coefficients is central to understanding how to expand expressions raised to a power, such as \((3m + 2)^4\). The binomial coefficients are represented by the notation \(\binom{n}{k}\), which is known as the binomial coefficient. This determines how many ways you can choose \(k\) objects from \(n\) total objects without regard to order.

These coefficients are calculated using the formula:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

where \(!\) indicates a factorial, meaning you multiply all positive integers up to the given number.

In the exercise, the expression \((3m + 2)^4\) has \(n = 4\). The coefficients \(1, 4, 6, 4, 1\) can be calculated directly using the formula, and they significantly affect the polynomial terms in the expansion process. Understanding these coefficients is crucial for simplifying the expanded form of any polynomial using the Binomial Theorem.

Pascal's Triangle

Pascal's Triangle is a powerful tool for quickly finding binomial coefficients for polynomial expansions. Each row in Pascal's Triangle corresponds to the coefficients needed for expanding binomials of increasing power.

To expand \((3m + 2)^4\), referring to the 4th row of Pascal's Triangle is essential because it gives the necessary coefficients \(1, 4, 6, 4, 1\). The triangle is built by beginning with a single \(1\) at the top, and each subsequent row begins and ends with \(1\). In between, each number is the sum of the two numbers directly above it from the previous row.

Pascal's Triangle not only simplifies the task of finding coefficients but also helps visualize patterns in binomial expansions. This orderly pattern allows for quick calculations and insights into complex polynomial expansions without having to manually compute each binomial coefficient using factorials.

Polynomial Expansion

Polynomial expansion using the Binomial Theorem involves transforming expressions like \((3m + 2)^4\) into a sum of terms. Each term consists of a binomial coefficient, a power of the first term, and a power of the second term. The essence of this expansion is to use the binomial coefficients to distribute the powers across the terms of the binomial expression.

In our example, the general term in the expansion of \((a + b)^n\) is given by:

• \(\binom{n}{k} a^{n-k} b^k\)

where \(a\) and \(b\) are the terms in the binomial, \(n\) is the power, and \(k\) is the specific term's position in the expansion (starting from 0).

The completed expansion yields:

• \(81m^4 + 72m^3 + 108m^2 + 64m + 16\).

By following this structured approach to polynomial expansion, the resulting polynomial expression delivers an accurate and simplified form of the original binomial, which can be used for further mathematical analysis or real-world applications.