Problem 41 - College Algebra

Question

In Exercises 39-48, find the term indicated in each expansion. $$(x-1)^{9} ;$$ fifth term

Step-by-Step Solution

Answer

The fifth term of the given expression \((x-1)^9\) is \(126x^5\).

Step 1: Identify the binomial coefficients for the fifth term

Using the Binomial theorem, the coefficient of the fifth term in the binomial expansion is given by \(\binom{n}{k}\), where n is the exponent of binomial and k is one less than the position of the term. Here, n = 9 and the fifth term implies k = 5 - 1 = 4.

Step 2: Compute the binomial coefficient

The binomial coefficient \(\binom{9}{4}\) calculates to 126.

Step 3: Find the fifth term

Now plug n=9, k=4 and the binomial coefficient into the Binomial theorem equation. The fifth term \( T_{5} \) is computed as \( T_{5} = \binom{9}{4} \cdot (x)^{9-4} \cdot (-1)^4 = 126 \cdot x^5. \)

Key Concepts

Binomial Theorem

Understanding the binomial theorem is essential for expanding algebraic expressions that are raised to a power. The binomial theorem provides a formula for expanding expressions of the form \( (a + b)^n \), where \( a \) and \( b \) are any numbers, and \( n \) is a positive integer. The expanded form is a sum of terms called binomial coefficients multiplied by \( a \) and \( b \) raised to varying powers.For example, when you have \( (x - 1)^9 \) and need to find the fifth term, the binomial theorem tells you that you will sum up different products of \( x \) and \( -1 \) raised to powers that add up to 9. The coefficients in front of each product are what gives the binomial expansion its unique values for each term. These coefficients can be found using combinations from Pascal's triangle or by direct calculation.

Binomial Coefficients

In the context of the binomial expansion, binomial coefficients are the numbers that appear as multipliers of the terms in the expanded binomial expression. They are a key part of Pascal's triangle and can be calculated using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( n! \) denotes the factorial of \( n \) and \( k \) corresponds to the specific term in the expansion. The factorial, denoted by an exclamation point, is the product of all positive integers up to that number.For instance, to find the coefficient of the fifth term in your example \( (x - 1)^9 \) as shown in the exercise, you would use \( k = 4 \) since the fifth term is obtained by using the value one less than the term's position. By plugging these into the formula, you get a coefficient of 126, which is then applied to the respective powers of \( x \) and \( -1 \) in the binomial expansion.

Algebraic Expressions

Algebraic expressions represent a combination of variables, numbers, and arithmetic operations. For example, in the binomial \( (x - 1)^9 \), \( x \) is the variable, and \( -1 \) is a constant. When dealing with binomial expansion, each term in the expanded form is an algebraic expression that shows the interplay between these variables and constants raised to different powers based on the expansion rule.In our example, the fifth term we found, \( 126 \cdot x^5 \), is an algebraic expression that demonstrates the product of the binomial coefficient (126) and the variable \( x \) raised to the fifth power. It's one of many terms that make up the entire expanded expression, which when combined, will include constants, variables, and various coefficients representing the expanded form of the original binomial.