Problem 41 - A Graphical Approach to College Algebra

Question

Write the indicated term of each binomial expansion. Fifteenth term of \(\left(a^{2}+b\right)^{22}\).

Step-by-Step Solution

Answer

The fifteenth term is \(319770 a^{16} b^{14}\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem allows us to expand expressions of the form \((x + y)^n\). The general term (k-th term) of the expansion is given by \(T_k = \binom{n}{k-1} x^{n-(k-1)} y^{k-1}\). Here, \(n\) is the exponent of the binomial, \(x\) and \(y\) are the terms of the binomial, and \(k\) is the term number.

Step 2: Identify Parameters

We need to find the 15th term of \((a^{2} + b)^{22}\). Here, \(n = 22\), \(x = a^2\), and \(y = b\). The term number \(k = 15\).

Step 3: Apply the General Term Formula

Using the formula for the k-th term, we have: \[ T_{15} = \binom{22}{15-1} (a^2)^{22-(15-1)} b^{15-1} \] Simplifying, \[ T_{15} = \binom{22}{14} (a^2)^8 b^{14} \]

Step 4: Simplify the Expression

Now, simplify \((a^2)^8\) as \(a^{16}\). The term becomes: \[ T_{15} = \binom{22}{14} a^{16} b^{14} \]

Step 5: Calculate the Binomial Coefficient

The binomial coefficient \(\binom{22}{14}\) can also be calculated using \(\binom{22}{14} = \binom{22}{8}\) which is \[ \binom{22}{8} = \frac{22 imes 21 imes 20 imes 19 imes 18 imes 17 imes 16 imes 15}{8 imes 7 imes 6 imes 5 imes 4 imes 3 imes 2 imes 1} = 319770 \] Thus, \(\binom{22}{14} = 319770\).

Step 6: Write the Final Term

The 15th term is therefore: \[ 319770 a^{16} b^{14} \].

Key Concepts

Binomial Theorem

The Binomial Theorem is an essential algebraic principle for expanding expressions raised to a power. When you see an expression like \((x + y)^n\),it might seem daunting to expand it directly especially for larger exponents. Fortunately, the Binomial Theorem gives us a systematic way to do this using a formula. Each term in the expansion can be identified through a specific formula: \(T_k = \binom{n}{k-1} x^{n-(k-1)} y^{k-1}\). In this context, \(n\) is the power of the binomial, \(x\) and \(y\) are the individual terms of the binomial, and \(k\) is the term number we are interested in.

• It helps in breaking down and calculating each individual term separately.
• This not only reduces the complexity but makes it easier to calculate specific terms without needing to fully expand the expression.

Learning how to apply the Binomial Theorem efficiently can be incredibly useful in various fields of maths and science, making complicated calculations more manageable.

Binomial Coefficient

A key part of the Binomial Theorem is the binomial coefficient, expressed as \(\binom{n}{k}\). This coefficient tells us how many ways we can choose \(k\) elements from \(n\) elements without regard to order. In the context of polynomial expansion, it represents the number multiplied by each term in the expansion. The formula to find a binomial coefficient is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]However, you don't always need to calculate factorials manually. Properties of binomial coefficients can simplify these calculations. For example, \(\binom{n}{k} = \binom{n}{n-k}\). This can significantly ease the computation by reducing the numbers involved in multiplication.

• The coefficient not only tells us the number of elements but also ensures the terms are expanded correctly.
• It's important in statistics, algebra, and probability.

Polynomial Expansion

Polynomial expansion refers to the process of expressing a binomial raised to a power as a polynomial—a sum of terms. Each term has a coefficient, a base component raised to a power, and possibly other elements. Using the Binomial Theorem, you can efficiently explore these expansions. For instance, in the binomial \((a^2 + b)^{22}\), we explore each individual term by computing the coefficient and determining the correct power of \(a^2\) and \(b\). The expansion begins with easier expressions and progressively identifies each term:\[T_k = \binom{22}{k-1} (a^2)^{22-(k-1)} b^{k-1}\]

• This structured exploration avoids the overwhelming task of multiplying the binomial manually.
• It often simplifies complex algebra into understandable pieces—crucial in algebraic problem-solving.

Once you master polynomial expansions using these methods, you can tackle more complex algebraic expressions with confidence and precision.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a particular value or relation. In context, an algebraic expression like \((a^2 + b)^{22}\) can be expanded using the Binomial Theorem into various terms.Understanding algebraic expressions is fundamental for higher mathematics. Recognizing each component within an expression allows for easier manipulation and transformation, as seen when we calculate specific terms in a binomial expansion.

• Variables like \(a\) and \(b\) stand for unknowns, making algebra flexible for various scenarios.
• Mastering how to rearrange, simplify, and expand algebraic expressions is integral for solving everything from simple equations to more complex functions.

Grasping these components makes learning algebra rewarding, as you're building the skills to solve real-world and theoretical problems.