Problem 35 - College Algebra

Question

In Exercises 31-38, write the first three terms in each binomial expansion, expressing the result in simplified form. $$\left(x^{2}+1\right)^{16}$$

Step-by-Step Solution

Answer

The first three terms of the binomial expansion of \( \left(x^{2} + 1\right)^{16} \) are \(x^{32}\), \(16x^{30}\), and \(120x^{28}\).

Step 1: Identify the Parameters

In expanding \( (a + b)^n \), identify \(a\), \(b\), and \(n\). In our case, \(a = x^2\), \(b = 1\), and the power \(n = 16\).

Step 2: First Term of the Expansion

By the binomial theorem, the first term in the expansion is \(a^n\). In this case, we compute \((x^{2})^{16}\) to obtain the first term, which simplifies to \(x^{32}\).

Step 3: Second Term of the Expansion

The second term in the binomial expansion is \(n a^{n-1}b\). So, replacing \(n\), \(a\) and \(b\) with 16, \(x^2\) and 1 respectively, we have \(16 * (x^2)^{15} * 1 = 16*x^{30}\).

Step 4: Third Term of the Expansion

The third term in the binomial expansion is \(\frac{n(n-1)}{2!} a^{n-2}b^2\). Replacing \(n\), \(a\) and \(b\) with 16, \(x^2\) and 1 respectively, and simplifying gives \(\frac{16*15}{2} * (x^2)^{14} * 1 = 120*x^{28}\).

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental principle in algebra that allows us to expand expressions raised to a power when the expression consists of a binomial, which is a polynomial with two terms. Typically, a binomial expression is in the form of \( (a + b)^n \), where \( a \) and \( b \) are any numbers or variables, and \( n \) is a positive integer.

The expansion is expressed as a sum involving terms of the form \( \frac{n!}{k!(n-k)!} a^{n-k}b^k \), where \( k \) ranges from 0 to \( n \) and \( n! \) denotes \( n \) factorial. This formula can generate each term of the expansion without directly multiplying the binomial by itself \( n \) times. The theorem helps to simplify expressions and solve algebraic problems that would be otherwise very tedious to calculate manually.

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial expression is \( x^2 + 5x + 6 \).

One of the most important properties of polynomial expressions is that they can be combined and manipulated using the distributive, associative, and commutative properties of multiplication and addition. These characteristics make polynomials a key subject of study in algebra and are integral in understanding binomial expansions, such as in the example of \( (x^2 + 1)^{16} \).

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form while preserving their original value. This process may include combining like terms, factoring, expanding expressions, and canceling common factors among others. In the context of binomial expansion, simplification is crucial for deriving the individual terms in the series and ultimately, for expressing the result concisely.

Let's take the second term from our example \( 16 \times (x^2)^{15} \). To simplify, we apply the rule of exponents to combine the powers and multiply the coefficients, leading to the simplified term \( 16 \times x^{30} \). Simplicity aids in comprehension and can significantly ease the process of solving more complex algebraic problems.

Exponentiation

Exponentiation is the mathematical operation, involving two numbers, where one number is raised to the power of another. This operation is represented as \( a^n \), where \( a \) is the base and \( n \) is the exponent or power. The exponent signifies how many times the base is multiplied by itself.

For example, in the binomial \( (x^2 + 1)^{16} \), each term of the expansion involves exponentiation. The first term, \( (x^{2})^{16} \), simplifies to \( x^{32} \), as you multiply the base \( x^2 \) by itself 16 times. Understanding the rules of exponents, such as \( (a^m)^n = a^{m \times n} \) and \( a^m \times a^n = a^{m+n} \) is key to handling operations in algebra involving powers and roots effectively.