Problem 53 - College Algebra with Corequisite Support

Question

Aaron wants to mulch his garden. His garden is \(x^{2}+18 x+81 \mathrm{ft}^{2}\). One bag of mulch covers \(x^{2}-81 \mathrm{ft}^{2}\). Divide the expressions and simplify to find how many bags of mulch Aaron needs to mulch his garden.

Step-by-Step Solution

Answer

Aaron needs \(\frac{x+9}{x-9}\) bags of mulch.

Step 1: Identify the mathematical expression

We need to divide the total area of Aaron's garden by the area one bag of mulch covers, resulting in \( \frac{x^2+18x+81}{x^2-81} \).

Step 2: Factor the expressions

Factor both the numerator and the denominator of the fraction. The numerator \(x^2+18x+81\) factors to \((x+9)^2\), and the denominator \(x^2-81\) factors to \((x-9)(x+9)\).

Step 3: Simplify the fraction

The fraction \(\frac{(x+9)^2}{(x-9)(x+9)}\) can be simplified by canceling out the common factor \((x+9)\). This gives \(\frac{x+9}{x-9}\).

Step 4: Evaluate the simplified expression

The simplified expression \( \frac{x+9}{x-9} \) indicates the number of bags, dependent on the value of \(x\). It's assumed this represents the solution unless any further specific instructions about \(x\) are provided.

Key Concepts

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler polynomial factors that, when multiplied together, yield the original polynomial. This is incredibly useful when simplifying expressions, solving equations, and performing polynomial division. In the context of the given exercise, we are working with two expressions that need to be factored to simplify the problem.

• The numerator expression is the quadratic polynomial \(x^2 + 18x + 81\). This is a perfect square trinomial and can be factored as \((x+9)^2\), meaning that both terms are identical.
• The denominator expression is another form, \(x^2 - 81\), which is a difference of squares. It factors into two binomials: \((x-9)(x+9)\). The format here is square root terms used with a positive and negative sign between them.

Factoring like this helps us see which parts of the expression can cancel out in division. Recognizing these forms quickly becomes second nature with practice. Whether dealing with trinomials like \((x+9)^2\) or difference of squares like \((x-9)(x+9)\), being adept at factoring is key to simplifying polynomial problems efficiently.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing a fraction consisting of polynomials into its simplest form. This is similar to simplifying numerical fractions, where the greatest common factor of the numerator and denominator is used to reduce the fraction. In this exercise, we simplified the rational expression \(\frac{(x+9)^2}{(x-9)(x+9)}\).

• First, we identify the common factor, which is \((x+9)\) in both the numerator and the denominator, enabling us to cancel out these terms.
• After cancelation, we are left with \(\frac{x+9}{x-9}\), a much simpler expression.

The goal is to make the expression as simple as possible, eliminating any redundant factors. It's important to remember that the value of the expression doesn't change during simplification, as long as no division by zero occurs. That's why recognizing common factors and factoring out appropriately are crucial steps in the process.

Quadratic Expressions

Quadratic expressions are polynomials of degree 2, usually written as \(ax^2 + bx + c\). They can appear in various mathematical contexts, from equations to expressions like in this exercise.

• In our example, both the numerator \(x^2 + 18x + 81\) and the denominator \(x^2 - 81\) are quadratic expressions.
• The standard form of a quadratic expression depicts them neatly for potential factoring opportunities or solving using the quadratic formula when rearranged as equations.

Quadratic expressions can represent areas and dimension problems, like calculating how much mulch you need for a garden space here. Understanding how to manipulate and simplify these expressions is a fundamental algebra skill that sets the stage for solving more complex mathematical problems. They provide a gentle introduction to the concepts of factoring and simplification that occur more widely in algebra.