Problem 60 - Beginning and Intermediate Algebra

Question

Find the indicated term of each binomial expansion. $$\left(2 y^{2}+z\right)^{10} ; \text { eighth term }$$

Step-by-Step Solution

Answer

The eighth term in the expansion of the given binomial expression, \((2y^2+z)^{10}\), is \(T_8 = 960y^{6}z^{7}\).

Step 1: Identify the components of the binomial expression

In this problem, we have the given binomial expression \((2y^2+z)^{10}\) and we need to find the eighth term in its expansion. The components of the binomial expression are: - a = \(2y^2\) - b = z - n = 10, which is the exponent.

Step 2: Understand the general term of a binomial expansion

The general term of a binomial expansion according to the binomial theorem is given by: \(T_r = C(n, k) * a^{n-k}*b^k\), where: - \(T_r\) is the r-th term in the expansion, - \(C(n, k)\) is the binomial coefficient (also denoted as \(\binom{n}{k}\) or nCk), - n is the exponent in the binomial expression, - a and b are the items of the binomial expression, and - k ranges from 0 to n. Since we need to find the eighth term, \(T_8\), we will use \(k = 7\) (as the first term corresponds to \(k = 0\)).

Step 3: Calculate the binomial coefficient

To find the binomial coefficient for the eighth term, we need to calculate: \(C(10, 7) = \binom{10}{7} = \frac{10!}{7!(3)!}\), where ! denotes the factorial of a number. We can calculate this as: \(C(10, 7)= \frac{10!}{7!3!} = \frac{10\times 9 \times 8}{3 \times 2 \times 1} = 10 \times 3 \times 4 = 120\).

Step 4: Apply the formula for the general term

Now, we can apply the formula for the eighth term, \(T_8\), using \(k = 7\): \(T_8 = C(10,7) * (2y^2)^{10-7} * z^7\) Plug in the binomial coefficient we found in step 3: \(T_8 = 120*(2y^2)^3*z^7\)

Step 5: Simplify the term

Now, simplify the term: \(T_8 = 120*(8y^6)*z^7\) \(T_8 = 960y^{6}z^{7}\) So, the eighth term in the expansion of the given binomial expression is: \(T_8 = 960y^{6}z^{7}\)

Key Concepts

Understanding the Binomial Theorem

The binomial theorem is a powerful algebraic tool used to expand expressions that are raised to a power, such as \( (a + b)^n \). This theorem provides a formula for expanding these expressions without multiply-ing them repeatedly. Each of the terms in the expansion is determined by the formula: \(T_k = C(n, k) \cdot a^{n-k} \cdot b^k\), where \(T_k\) represents a specific term in the expansion. This equation helps identify the terms systematically, especially for large powers.

Core components of the binomial theorem include:

• Each expansion involves terms formed by varying powers of the two variables, here labeled \(a\) and \(b\).
• The coefficients of these terms are determined by binomial coefficients, which we will explore further in the next section.
• The power of \(a\) starts high and decreases, while the power of \(b\) starts at 0 and increases as you move through the terms.

For instance, for \( (2y^2 + z)^{10} \), to find the eighth term, one must understand the role of the binomial theorem in systematically locating and calculating specific terms within the expanded polynomial.

Exploring Binomial Coefficients

Binomial coefficients play a crucial role in the binomial theorem. They dictate the weight or multiplicative factor of each term in the expansion. The binomial coefficient \(C(n, k)\), often expressed as \( \binom{n}{k} \), reveals the number of ways to pick \(k\) elements from a total of \(n\) elements without regard to the order.

The calculation of a binomial coefficient is rooted in combinations from combinatorics and is given by:
\[ C(n, k) = \frac{n!}{k!(n-k)!} \]
Here, \(!\) denotes the factorial operation, which is illustrated in the next section.

• In the exercise, for the eighth term, we use \( C(10, 7) \), highlighting that \( k = 7 \) corresponds to the exponents of each term's components being chosen.
• This results in an expanded form that systematically grows the power of \(b\) from 0 at the first term, upwards, as the power of \(a\) decreases.

Understanding these coefficients not only simplifies polynomial expansion processes but also provides a deeper insight into pattern recognition within mathematical series.

Factorial Calculation Explained

Factorials are foundational in calculating many mathematical expressions, particularly binomial coefficients. The factorial of a non-negative integer \(n\), denoted as \(n!\), is the product of all positive integers less than or equal to \(n\). Factorials are a key element in combination calculations, as well.

For example:

• \(0!\) is defined to be \(1\).
• \(5!\) or "5 factorial" is \(5 \times 4 \times 3 \times 2 \times 1 = 120\).

In our binomial coefficient calculation, \( C(10,7) = \frac{10!}{7!3!} \) requires computing:

• \(10!\) which involves the multiplication of integers from \1\ to \10\.
• Furthermore, dividing as indicated by the formula simplifies the calculation into manageable steps.

Understanding and calculating with factorials is crucial for accurately determining the terms in a binomial expansion, revealing the structure and symmetry of polynomial expansions.