Question

MODELING WITH MATHEMATICS A farmer plants a garden that contains corn and pumpkins. The total area (in square feet) of the garden is modeled by the expression \(2 x^2+5 x+4\). The area of the corn is modeled by the expression \(x^2-3 x+2\). Write an expression that models the area of the pumpkins.

Short answer

The expression that models the area of the pumpkins is \(x^2 + 8x + 2\).

Step-by-step solution

Understand the problem

First, ensure that the given expressions makes sense: the total area of the farm is \(2x^2+5x+4\) and the area taken by corn is \(x^2-3x+2\). The aim is to find the area taken by pumpkins. According to the problem, the total area is made up by the area taken up by corn and pumpkins. Therefore, we can find the pumpkin area by subtracting the corn's area from the total area.

Subtraction of expressions

In this step, we subtract the corn's area expression (\(x^2-3x+2\)) from the total area expression (\(2x^2+5x+4\)). This procedure involves subtracting the coefficients of like terms (terms with the same power): subtract the coefficients of \(x^2\), \(x\), and the constant terms.

Solve the expression

The solution is the resultant expression after performing the subtraction: \(2x^2+5x+4 - (x^2-3x+2) = x^2 + 8x + 2\)

Key concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as addition, subtraction, multiplication, and division). Unlike an equation, an algebraic expression doesn’t have to include an equals sign. It’s essentially a way of writing out a problem or situation in mathematical terms.

For example, in our garden problem, the total area of the garden is represented by the algebraic expression 2x^2 + 5x + 4, where x could represent a unit of measurement that scales with the size of the garden. The power of algebraic expressions lies in their ability to model real-world situations, such as the farmer’s garden, and to manipulate these expressions according to the rules of algebra to solve problems.

Subtraction of Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, structured across terms of varying degrees. Subtracting polynomials involves combining like terms, which are terms that have the same variables raised to the same power. The subtraction process is straightforward:

• Distribute any negative signs through the parentheses.
• Combine like terms by subtracting their coefficients.

Considering our garden example, to find the area of the pumpkins, we subtract the area of the corn (x^2 - 3x + 2) from the total garden area (2x^2 + 5x + 4), resulting in a new polynomial which represents the area left for the pumpkins. The subtraction process demonstrates an essential skill in algebra: dealing with multiple polynomial terms and simplifying them to find a solution.

Problem-solving in Algebra

Problem-solving in algebra often involves interpreting real-world situations and representing them with algebraic expressions, then using the properties and rules of algebra to find an unknown quantity. In our farming scenario, the problem was approached systematically:

First, we understood what was asked: the separate areas of corn and pumpkins making up the total area of a garden. Next, we identified the algebraic expressions representing those areas. Then, we applied the operation of subtraction to these expressions, all the while working methodically to ensure each term was correctly combined.

Critical Steps in Problem Solving

These include:

• Understanding the problem and what is being asked.
• Breaking the problem into manageable parts.
• Choosing the correct method to use for the solution (in this case, subtraction of polynomials).
• Performing calculations carefully and accurately.
• Checking your solution to ensure it makes sense in the context of the problem.

This approach isn't just applicable to our farmer's garden but to any problem where algebra is a tool for finding unknowns in equations and expressions.