Problem 74 - Algebra 2

Question

Explain how you can use the Binomial Theorem to find the sixth term in the expansion of \((2 x-3 y)^{7} .\)

Step-by-Step Solution

Answer

Hence, the sixth term in the expansion of \((2x - 3y)^7\) is \( \binom{7}{5} (2x)^2 (-3y)^5 \) where \( \binom{7}{5} \) is the binomial coefficient calculated using the formula \( \frac{n!}{r!(n-r)!} \), and \( (2x)^2 \) and \( (-3y)^5 \) are the powers of \( a \) and \( b \) respectively.

Step 1: Understanding the Problem

The goal is to find the sixth term in the binomial expansion. According to the binomial theorem, the 'r'-th term of a binomial expansion \( (a+b)^n \) is given by \( \binom{n}{r-1} a^{n-r+1} b^{r-1} \). This form is applicable here as the 6th term corresponds to \( r = 6 \). The coefficients \(\binom{n}{r-1}\) are called binomial coefficients and can be calculated using factorials.

Step 2: Calculate the binomial coefficient

First, calculate the binomial coefficient of the 6th term, considering that in this case \( n = 7 \) and \( r = 6 \). So, calculate \( \binom{7}{6-1} = \binom{7}{5} \), using the formula for binomial coefficients which is \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), where '!' denotes the factorial of a number.

Step 3: Find the powers for \(a\) and \(b\)

Next, we need to find the power of \(a\) and \(b\) in the 6th term. These can be represented by \(a^{n-r+1}\) and \(b^{r-1}\) respectively. From the binomial expression \((2x-3y)^7\), \(a=2x\) and \(b= -3y\). Substituting \(n = 7\), \(r = 6\), \(a = 2x\) and \(b = -3y\), we get \( a^{n-r+1} = (2x)^{7-6+1} = (2x)^2\) and \(b^{r-1} = (-3y)^{6-1} = (-3y)^5\)

Step 4: Calculate the 6th term

Finally, using the formula for the \( r \)-th term of the binomial expansion \( \binom{n}{r-1} a^{n-r+1} b^{r-1} \), substitute the binomial coefficient \( \binom{7}{5} \), \( a^{n-r+1} = (2x)^2 \), and \( b^{r-1} = (-3y)^5 \) to obtain the 6th term.

Key Concepts

Binomial Expansion

The concept of binomial expansion is a powerful tool in algebra, particularly when dealing with expressions raised to a power. Essentially, the binomial theorem provides us with a way to expand expressions of the form \((a + b)^n\) into a sum involving terms of the forms \(a^k b^{n-k}\). By using the theorem, we can systematically expand and simplify binomial expressions.

• Each term in the expansion is called a binomial term.
• The exponents of \(a\) and \(b\) add up to \(n\), the power of the binomial expression.

For example, if you have \((2x - 3y)^7\) and want to find a specific term, like the sixth term, the binomial theorem guides you to calculate it directly, without fully expanding the whole expression.

Binomial Coefficients

Binomial coefficients are the numbers that appear in the binomial expansion formula, and they play a significant role in determining the terms of the expansion. You might recognize them from Pascal's Triangle, where each number is the sum of the two numbers directly above it.

• The binomial coefficient for a particular term in the expansion is given by \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\).
• Here, \(n!\) (n factorial) is the product of all positive integers less than or equal to \(n\).
• These coefficients are crucial for ensuring the terms in the binomial expansion are properly weighted.

For the sixth term of \((2x - 3y)^7\), the coefficient is \(\binom{7}{5}\), which can be calculated using the factorial formula mentioned above.

Factorials

Factorials are a fundamental mathematical concept essential for calculating permutations, combinations, and notably, binomial coefficients. The factorial of a number \(n\), denoted as \(n!\), is calculated as the product \(n \times (n-1) \times (n-2) \times ... \times 2 \times 1\).

• For instance, \(5!\) equals \(5 \times 4 \times 3 \times 2 \times 1 = 120\).
• Factorials grow very quickly and are used within the formula for binomial coefficients \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\).

In the previous example regarding the expansion of \((2x - 3y)^7\), we calculated \(\binom{7}{5}\) using factorial values, demonstrating the practical utility of factorials in binomial expansions.

Powers of Expressions

Understanding powers of expressions is key to working with algebraic expansions, such as those seen in the binomial theorem. When we talk about powers, we refer to multiplying a base number by itself a certain number of times.

• For example, \((a^2)\) means \(a \times a\).
• Powers are used to determine how many times an expression, such as a variable or a polynomial term, should be multiplied by itself.

In the case of \((2x - 3y)^7\), calculating the powers of \(a\) and \(b\) for a specific term involves raising \(a=2x\) and \(b=-3y\) to the powers determined by the binomial formula, specifically \((2x)^2\) and \((-3y)^5\). This process allows for computing the various terms and coefficients that make up the complete expression.