Problem 40 - A Graphical Approach to Precalculus with Limits

Question

Write the indicated tem of each binomial expansion. Eighth term of \((2 c-3 d)^{14}\)

Step-by-Step Solution

Answer

The eighth term is \(-960180480 c^7 d^7\).

Step 1: Identify the Term Formula

For a binomial expansion, the term we want can be found using the following formula:\[ T_{k+1} = \binom{n}{k} a^{n-k} b^k \]where \( \binom{n}{k} \) represents the binomial coefficient, \( a \) and \( b \) are the terms of the binomial \((a + b)^n\), \( n \) is the degree of the binomial, and \( k \) is the term position minus one.

Step 2: Assign Values

In the expression \((2c - 3d)^{14}\), we identify:- \(n = 14\)- \(a = 2c\)- \(b = -3d\)We are finding the eighth term, so \(k+1 = 8\), thus \(k = 7\).

Step 3: Calculate the Binomial Coefficient

Calculate \( \binom{14}{7} \) as follows:\[\binom{14}{7} = \frac{14!}{7!(14-7)!} = \frac{14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 3432\]

Step 4: Substitute Values into the Term Formula

With \(k = 7\), substitute the values into the binomial term formula:\[T_8 = \binom{14}{7} (2c)^{14-7} (-3d)^7\]

Step 5: Simplify the Expression

Calculate the powers and simplify:- \((2c)^7 = 2^7 c^7 = 128c^7\)- \((-3d)^7 = (-1)^7 \cdot 3^7 d^7 = -2187d^7\)Combine these results:\[T_8 = 3432 \cdot 128c^7 \cdot (-2187d^7)= -960180480 c^7 d^7\]

Step 6: Conclude the Calculation

The eighth term of the binomial expansion is:\[-960180480 c^7 d^7\]

Key Concepts

Binomial Coefficient

In the expansion of a binomial expression such as \((a + b)^n\), the binomial coefficient is a crucial element that helps us find specific terms in the expansion. The binomial coefficient, denoted as \(\binom{n}{k}\), indicates the number of ways to choose \(k\) items from a total of \(n\) without regard to order.
It can be calculated using the formula:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

• \(n!\) (read as "n factorial") is the product of all positive integers up to \(n\).
• For example, in our problem, we calculated \(\binom{14}{7}\) which equals to 3432, meaning there are 3432 different ways to reach the eighth term using combinations.

Term Formula

The term formula in a binomial expansion allows us to find any term within the expansion quickly without having to expand the whole polynomial. The formula is given by:
\[T_{k+1} = \binom{n}{k} a^{n-k} b^k\]
This formula incorporates several components:

• \(\binom{n}{k}\) which is the binomial coefficient we've discussed.
• \(a^{n-k}\) and \(b^k\) are the components raised to specific powers making up part of the term.

In our example, to find the eighth term, we determined \(k\) as 7 (since the term's position is \(k+1\)), then plugged in these values allowing for straightforward calculation.

Polynomial Degree

In binomial expressions, the degree of the polynomial is represented by \(n\) in the expression \((a + b)^n\). The degree indicates the highest power present when the binomial is expanded, resulting in \(n+1\) terms.

• In our exercise with \((2c - 3d)^{14}\), 14 is the degree.
• The degree influences the binomial coefficient, affecting the weight or prominence each term in the expansion has.

Understanding the degree helps in grasping the overall structure of the expansion, knowing how many terms it will have in total.

Exponent Calculation

Calculating exponents is a critical part of simplifying terms in any polynomial expression, including binomial expansions. Here's how it works:

• For any term in the expansion, calculate \(a^{n-k}\). For example, in step 5 we computed \((2c)^7\) to get \(128c^7\).
• Next, calculate \(b^k\). In the solution, \((-3d)^7\) was calculated as \(-2187d^7\).

When combining these results, consider the signs, as negative values can affect the final outcome. Like in our exercise, even powers keep signs the same, but odd powers (like \((-3d)^7\)) will switch negative signs. Proper exponent calculation ensures each term is correctly simplified in the expansion.