Problem 83 - Algebra for College Students

Question

Find the indicated products. Assume all variables that appear as exponents represent positive integers. $$\left(2 x^{n}+5\right)^{2}$$

Step-by-Step Solution

Answer

\((2x^n + 5)^2 = 4x^{2n} + 20x^n + 25\).

Step 1: Identify the Expression

We need to find the product of \((2x^n + 5)^2\). This is a binomial squared, which means we will apply the formula \((a + b)^2 = a^2 + 2ab + b^2\) to expand it.

Step 2: Apply the Binomial Squares Formula

For the expression \((2x^n + 5)^2\), let \(a = 2x^n\) and \(b = 5\). Applying the formula \((a + b)^2 = a^2 + 2ab + b^2\), we calculate each term separately.

Step 3: Calculate Each Term

1. Calculate \(a^2 = (2x^n)^2 = 4(x^n)^2 = 4x^{2n}\).2. Calculate \(2ab = 2 \cdot 2x^n \cdot 5 = 20x^n\).3. Calculate \(b^2 = 5^2 = 25\).

Step 4: Write the Final Expanded Expression

Combine all the terms to get the final expanded expression. Thus, \((2x^n + 5)^2 = 4x^{2n} + 20x^n + 25\).

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of the same number or variable. When you see a number like \( x^n \), it means that \( x \) is multiplied by itself \( n \) times.

• The base \( x \) is the number being multiplied.
• The exponent \( n \) indicates how many times the base is used as a factor.
• Exponents can be positive, negative, or even zero.
• In our example, \((2x^n)^2\) involves an exponent of 2 applied to the term \(2x^n\).

Calculating \((2x^n)^2\), we first square the coefficient 2 (resulting in 4) and apply the exponent rule that states \((x^n)^2 = x^{2n}\), leading to \(4x^{2n}\). Understanding these rules allows for simplifying expressions and solving complex algebraic problems.

Polynomial Expansion

Polynomial expansion involves expressing a polynomial raised to a power in its expanded form. It typically uses methods like the distributive property or special formulas.

• When expanding \((a + b)^n\), you aim to find an equivalent expression in a fully expanded form.
• This often involves using binomial expansion, which is particularly useful for dealing with expressions like \((2x^n + 5)^2\).
• In this exercise, we use the formula for binomial squares: \((a + b)^2 = a^2 + 2ab + b^2\).

Applying this formula efficiently turns a complex polynomial into manageable parts. Thus, the expression \((2x^n + 5)^2\) is expanded by calculating each component as \(a^2\), \(2ab\), and \(b^2\), then summing them to get the final result.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. They form the building blocks of algebra, often used to model real-world problems.

• An algebraic expression does not have an equality sign, distinguishing it from equations.
• Expressions can include constants, such as 5, or variables with exponents, such as \(x^n\).
• The polynomial \(2x^n + 5\) is an example, where \(2x^n\) is a term with a variable part and 5 is a constant term.

Simplifying and manipulating these expressions, like in the binomial expansion of \((2x^n + 5)^2\), allows us to break down complex problems into organized steps, facilitating understanding and solution of high-level algebra.