Problem 46

Question

Find the specified term. The second term of $(m-n)^{9}$

Step-by-Step Solution

Verified

Answer

The second term is $-9m^8n$.

1Step 1: Understand the Binomial Theorem Formula

The binomial theorem provides a way to expand binomials raised to a power. The formula is: $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$. Here, $a = m$, $b = -n$, and $n = 9$.

2Step 2: Identify the Formula for the Specific Term

The $k^{th}$ term in the expansion using the binomial theorem is given by $T_{k+1} = \binom{n}{k} a^{n-k} b^k$. Here we need the second term, so $k = 1$.

3Step 3: Apply Values to the Formula

Plug $k = 1$ and the known values into the formula: $T_2 = \binom{9}{1} (m)^{9-1} (-n)^1$.

4Step 4: Compute the Binomial Coefficient

Calculate the binomial coefficient: $\binom{9}{1} = 9$.

5Step 5: Simplify the Term Expression

Insert the computed coefficient into the term expression: $T_2 = 9 \cdot m^8 \cdot (-n)$. Simplify: $T_2 = -9m^8n$.

Key Concepts

Binomial ExpansionBinomial CoefficientsAlgebraic Expressions

Binomial Expansion

The Binomial Expansion is a powerful method that simplifies the process of expanding expressions that are raised to a power. In mathematics, binomials are expressions composed of two terms, like

$(a + b)$.

To understand this expansion, think of it as breaking down large powers into a sum of smaller parts, each part being a product of powers of the individual terms. The expanded version of a binomial like

$(m - n)^9$

can be expressed as a series of terms using a specific pattern provided by the binomial theorem. In essence, what this pattern involves is applying the binomial theorem formula:

\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\].

By using a sum notation, we can easily derive each term in the expansion without manually multiplying each part of the binomial which saves time and reduces complexity. In this case, $a = m$, $b = -n$, and $n = 9$.

Binomial Coefficients

Binomial coefficients are a crucial component of the binomial expansion. These coefficients essentially tell us how many different ways we can pick elements from a set, and they are important in calculating each term in the expanded form of a binomial. The coefficient for each term is found using the formula:

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

where $n!$ ("n factorial") stands for the product of all positive integers up to $n$, and $k!$ is the product of all integers up to $k$.These coefficients are symmetric; this means that the first coefficient is the same as the last, the second the same as the second last, and so forth. For the task at hand, we're looking at a binomial expansion of degree 9 where we need the second term ($k=1$).Plugging our values into the formula delivers $\binom{9}{1} = 9$, instructing us to multiply the other pieces of this term calculation by 9 to form part of the full expansion.

Algebraic Expressions

An algebraic expression involves numbers, variables, and operations combined in a meaningful way. When it comes to binomial expansion, these expressions enable us to translate simple variable-containing brackets into more complex polynomial forms. For example, an expression like

$T_2 = 9 \cdot m^8 \cdot(-n)$

is derived from utilizing the binomial formula. Here each part—$9$, $m^8$, and $-n$—are results of applying the respective terms and coefficients from the binomial theorem. These expressions make it easier to visualize and simplify problems like finding a specific term within a polynomial expansion.Using algebraic techniques like substitution can bring further simplification and better understanding of binomials when we apply known values or conditions to find specific solutions—for instance, recognizing