Problem 51 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ (x+1)^{3} $$

Step-by-Step Solution

Answer

The expansion of \((x+1)^3\) is \(x^3 + 3x^2 + 3x + 1\).

Step 1: Identify a, b and n values

First identify the values of \(a\), \(b\), and \(n\). In this case, \(a = x\), \(b = 1\), \(n = 3\).

Step 2: Apply the Binomial theorem

Next, apply the binomial theorem to the equation. This states that \((a+b)^{n} = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\). Given our values, our function simplifies becomes \(x^3 + 3x^2*1 + 3x*1^2 + 1^3\).

Step 3: Simplify the expression using arithmetic

In the final step, simplify the result from Step 2 using basic arithmetic: \(x^3 +3x^2 + 3x + 1\).

Key Concepts

Polynomial Expansion

Polynomial expansion refers to the process of expressing a power of a binomial, like \((x+1)^{3}\), as a sum of terms. This concept is crucial in algebra as it turns complicated expressions into a set of simpler terms added together. When expanding a binomial, the Binomial Theorem is often employed. This theorem allows us to expand any expression of the form \((a+b)^n\) efficiently.
\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
This equation uses the concept of binomial coefficients, represented by \( \binom{n}{k} \), which calculates the number of ways to choose \(k\) elements from \(n\). Each term in the expanded polynomial combines these coefficients with the respective powers of \(a\) and \(b\). This structured approach simplifies large powers of binomials into a manageable series of terms.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and operators like addition and multiplication. They form the building blocks of algebra and are utilized to formulate mathematical statements about quantities and their relationships.
In the expression \((x+1)^3\), \(x\) is a variable and \(1\) is a constant. Algebra enables us to manipulate these symbols according to defined rules to solve equations and inequalities. By using algebraic properties, such as the distributive law and the Binomial Theorem, we can transform \((x+1)^3\) from its compact form into an expanded polynomial \(x^3 + 3x^2 + 3x + 1\).
Understanding the structure of algebraic expressions provides a strong foundation for solving more complex algebraic equations and understanding the relationships between different quantities.

Exponentiation

Exponentiation is a mathematical operation involving two numbers: a base and an exponent. It represents repeated multiplication of the base. For example, \((x+1)^3\) means \((x+1)\) multiplied by itself three times. This operation is a cornerstone of higher-level mathematics.
With exponentiation, expressions can be simplified and manipulated easily. The Binomial Theorem shows the power of exponentiation by providing a method to handle expressions raised to a power. It uses the formula
\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
to transform the repeated multiplication into a sum of terms. Each term is comprised of a combination of the base's components, powered and multiplied by the coefficients derived from the theorem.
Exponentiation is fundamental not only in pure algebra but also in various scientific applications, demonstrating its wide-ranging importance across multiple disciplines.