Problem 85 - Elementary Algebra

Question

A square is a rectangle in which the lengths of all four sides are equal. If \(s=\) length of a side, write a polynomial expression in \(s\) that represents the perimeter.

Step-by-Step Solution

Answer

The polynomial expression for the perimeter of the square is \(4s\).

Step 1: Understanding the Perimeter of a Square

The perimeter of a square is the total distance around the outside of the square. Since a square has four equal sides, the perimeter is calculated by adding up the lengths of all four sides.

Step 2: Expressing Perimeter in Terms of Side Length

If each side of the square is of length \(s\), then the perimeter is given by the sum of the lengths of all four sides. This can be expressed as: \text{Perimeter} = s + s + s + s\.

Step 3: Simplifying the Expression

Combine the terms to simplify the expression. Since there are four sides each of length \(s\), the expression simplifies to: \text{Perimeter} = 4s\.

Step 4: Write the Polynomial Expression

The simplified expression \(4s\) is already a polynomial in one variable. So, the polynomial expression for the perimeter of the square in terms of \(s\) is \4s\.

Key Concepts

Algebraic Expressions

Algebraic expressions are a way to represent numbers and operations using symbols and variables. In the given exercise, we use the variable \( s \) to represent the length of the side of the square. An algebraic expression can combine variables, coefficients, and mathematical operations.

For example, if we write 4s, it means multiplying the variable s by 4. Algebraic expressions are useful because they allow us to generalize mathematical concepts and solve problems more easily. By representing the perimeter of the square as 4s, we can quickly calculate the perimeter for any given side length.

Polynomial

A polynomial is an algebraic expression that includes terms made up of a variable raised to a non-negative integer power and multiplied by a coefficient. The expression 4s from the exercise is an example of a polynomial.

In general, a polynomial can have multiple terms, like \( ax^n + bx^{n-1} + \ldots + c \). But in our case, we have only one term, which is a very simple polynomial called a monomial. The term 4s means that the perimeter is dependent on the length of the side and that we multiply it by 4.

Polynomials can be more complex, but this exercise keeps it straightforward with just one variable and one term.

Geometry

Geometry deals with the properties and measurements of shapes and spaces. In this exercise, we're focused on the geometric shape known as a square. A square has four equal sides and four right angles.

To find the perimeter, we sum the length of all four sides. Understanding this basic geometry concept helps us see why the algebraic expression for the perimeter is 4s. Geometry gives us the framework to visualize and calculate lengths, areas, and volumes, making it an essential part of math.

When you know that a square's sides are equal, you can easily compute its perimeter, area, and even the length of its diagonals. This exercise links geometric understanding with algebraic expressions.