Problem 44 - A Graphical Approach to Precalculus with Limits

Question

Write the indicated tem of each binomial expansion. Tenth term of \(\left(a^{3}+3 b\right)^{11}\)

Step-by-Step Solution

Answer

The tenth term is \(1082565 a^6 b^9\).

Step 1: Identify the General Term Formula

The general term (the k-th term) of the binomial expansion \((x + y)^n\) is given by the formula: \[T_{k} = \binom{n}{k-1} x^{n-k+1} y^{k-1}\]For our expression \((a^3 + 3b)^{11}\), the formula becomes:\[T_{k} = \binom{11}{k-1} (a^3)^{11-k+1} (3b)^{k-1}\]

Step 2: Determine the Position of the Tenth Term

We want the tenth term of the expansion. In the binomial expansion formula, the first term corresponds to \(k=1\). Therefore, the tenth term corresponds to \(k=10\).

Step 3: Apply the General Term Formula for k=10

Substitute \(k=10\) into the formula:\[T_{10} = \binom{11}{9} (a^3)^{11-10+1} (3b)^{10-1}\]Simplify the exponents:\[(a^3)^{2} = a^{6}\]\[(3b)^{9} = 19683b^9\]

Step 4: Calculate the Binomial Coefficient and Simplify

Calculate the binomial coefficient:\[\binom{11}{9} = \binom{11}{2} = 55\]Thus, the term becomes:\[T_{10} = 55(a^6)(19683b^9)\]

Step 5: Simplify the Expression

Multiply the coefficients together:\[55 \times 19683 = 1082565\]The tenth term is therefore:\[T_{10} = 1082565 a^6 b^9\]

Key Concepts

Binomial Theorem

The Binomial Theorem is a fundamental principle in algebra that allows us to expand expressions of the form \((x + y)^n\). It provides a systematic method to write out the long multiplication of binomial expressions without direct multiplication.

The theorem states that any power of a binomial expression \((x + y)^n\) can be expressed as a sum of terms of the form \(\binom{n}{k} x^{n-k} y^k\). These components include:

• \(\binom{n}{k}\): The binomial coefficient, representing a specific term's combinatorial selection, calculated as \(\frac{n!}{k!(n-k)!}\).
• \(x^{n-k}\): The power of the first term, decreasing with each subsequent term.
• \(y^k\): The power of the second term, increasing with each subsequent term.

This theorem is instrumental in various applications, including polynomial expansions, probability, and series calculations, making it an essential tool for students to master.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. It's a unifying thread of almost all mathematics. In the context of binomial expansions, algebra involves understanding how to apply these rules to simplify and solve expressions.

Here's why algebra is crucial for binomial expansions:

• It teaches the use of variables and constants, essential in forming algebraic expressions such as \((a^3 + 3b)^{11}\).
• It includes operations on expressions, such as addition, subtraction, and multiplication, allowing for simplifications.
• It develops problem-solving skills crucial for determining the specific terms in an expansion, such as the tenth term in our original problem.

It is a fundamental area of mathematics, serving as a foundation for advanced studies. Understanding algebraic principles is essential for tackling complex polynomial equations effectively.

Exponentiation

Exponentiation refers to the operation of raising a number, known as the base, to a power, which indicates how many times the base is multiplied by itself. In binomial expansions, exponentiation is repeatedly implemented. For example, in \((a^3)^{11-k+1}\), exponentiation illustrates the repetition in processing.

Key points about exponentiation in binomial expansions include:

• Understanding base powers: \(a^3\) and how this base is manipulated across terms.
• The rule \((a^m)^n = a^{m \cdot n}\), essential for simplifying expressions such as \((a^3)^2\).
• Application of the power rule in the context of binomial expansion, ensuring accurate computations of terms like \(a^6\).

Mastering exponentiation is crucial for simplifying binomial expansions and deriving accurate results.

Polynomial Expansion

Polynomial Expansion involves expressing a polynomial in an extended form. Binomial expansions are specific types of polynomial expansions. They express powers of a sum in terms of a sum of powers. Polynomial formulations like these simplify otherwise tedious multiplicative processes.

When expanding binomials:

• Terms are expressed as a summation of products, each combination derived using the binomial theorem.
• Coefficients are calculated using combinatorial mathematics to determine the number of ways each term can be selected.
• Gradually increasing/decreasing powers of individual terms are identified and computed systematically.

Polynomial expansion is not only vital in pure mathematics but also in fields like physics and engineering, where mathematical modeling and expansions are often required.