Problem 3 - Elementary Algebra

Question

How many \(y^{2}(b+2) ' s\) in \(8 x y^{2}(b+2)(b-6) ?\)

Step-by-Step Solution

Answer

Answer: There are \(8x(b-6)\) \(y^2(b+2)\)'s in the expression \(8xy^2(b+2)(b-6)\).

Step 1: Write down the given expression and the divisor

We have the expression: \(8xy^2(b+2)(b-6)\) and we want to divide it by \(y^2(b+2)\).

Step 2: Divide the expression by the divisor

Now we will divide the given expression by the divisor: \(\frac{8xy^2(b+2)(b-6)}{y^2(b+2)}\)

Step 3: Cancel the common terms

We can see that there are common terms in the expression and the divisor: \(\frac{8xy^2\cancel{(b+2)}(b-6)}{\cancel{y^2}\cancel{(b+2)}}\)

Step 4: Write the simplified expression

After cancelling out the common terms, the simplified expression is: \(8x(b-6)\) There are \(8x(b-6)\) \(y^2(b+2)\)'s in the expression \(8xy^2(b+2)(b-6)\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are a fundamental concept in algebra and serve as a way to model real-world problems using numbers and symbols. They consist of variables, constants, and arithmetic operations. For instance, the expression \(8xy^2(b+2)(b-6)\) includes variables like \(x\), \(y\), and \(b\), as well as the constant \(8\). Variables represent unknown values, whereas constants are fixed numbers. Operations like multiplication combine these variables and constants to form terms. In this expression, you also see grouping symbols, such as parentheses, which indicate the order in which operations should be performed. Algebraic expressions can become complex, but breaking them down into their individual components helps make them manageable.

The Process of Simplification

Simplification is the process of reducing an algebraic expression to its simplest form. It involves combining like terms, canceling common factors, and performing the arithmetic indicated by operations. In the exercise provided, after identifying the expression \(8xy^2(b+2)(b-6)\) and the divisor \(y^2(b+2)\), simplification begins by dividing the expression by its divisor. The process involves these steps:

• Identify and cancel common terms in the numerator and the denominator, which are \(y^2\) and \(b+2\) in this case.
• The remaining expression after cancellation is simplified to \(8x(b-6)\), which is a shorter and more manageable form of the original expression.

Simplifying algebraic expressions not only makes them easier to understand but also easier to work with in further calculations.

Role of Common Factors in Simplification

Identifying and eliminating common factors is a pivotal aspect of simplifying algebraic expressions. A common factor is a term that appears in both the numerator and the denominator or within separate terms of an expression. In the given problem, \(y^2(b+2)\) stood as common factors in both the numerator \(8xy^2(b+2)(b-6)\) and the divisor. By canceling these, the expression is reduced in complexity. This process ensures that you focus only on the unique parts of the expression, often making subsequent calculations less prone to error. Common factors are best identified by:

• Careful inspection of each term and partitioning them into smaller multiplicative components.
• Using visual aids, like writing each term separately, helps in seeing these common factors clearly.
• Practicing with various problems enhances the ability to quickly spot and utilize common factors.

Handling common factors efficiently streamlines solving algebraic equations and assists in achieving more accurate results.