Problem 45 - Algebra 1

Question

Which product equals the quotient \((2 x+2) \div \frac{x^{2}+x}{4} ?\) (A) \(\frac{1}{2 x+2} \cdot \frac{x^{2}+x}{4}\) (B) \(\frac{2 x+2}{1} \cdot \frac{x^{2}+x}{4}\) (C) \(\frac{1}{2 x+2} \cdot \frac{4}{x^{2}+x}\) (D) \(\frac{2 x+2}{1} \cdot \frac{4}{x^{2}+x}\) (E) \(\frac{2 x+2}{2 x+2} \cdot \frac{4}{x^{2}+x}\)

Step-by-Step Solution

Answer

The quotient \((2 x+2) \div \frac{x^{2}+x}{4}\) is equivalent to the product (C) \(\frac{1}{2 x+2} \cdot \frac{4}{x^{2}+x}\).

Step 1: Simplify the Dividend Expression

Simplify the dividend expression (2x+2). This can be expressed as 2*(x+1).

Step 2: Simplify the Divisor Expression

Simplify the divisor expression \(\frac{x^2+x}{4}\) by factoring out x, resulting in \(\frac{x*(x+1)}{4}\).

Step 3: Change Division to Multiplication

Change the equation from division to multiplication, since division by a fraction is the same as multiplication by its reciprocal. We then get \(2*(x+1)*\frac{4}{x*(x+1)}\).

Step 4: Simplify the Resulting Expression

Simplify the final expression. The (x+1) in the numerator and the denominator cancel out due to the property of equivalence. The resulting expression is \(2*\frac{4}{x} = \frac{8}{x} \).

Step 5: Identify the Corresponding Product

Identify the product option which corresponds to the simplified expression. From the given options this is (C) \(\frac{1}{2x+2} * \frac{4}{x^2+x}\).

Key Concepts

Simplification of Algebraic Expressions

Algebraic expressions can often seem complicated, but simplification involves breaking them down into more manageable parts. This is similar to cleaning up sentences in a cluttered paragraph to make them more understandable. By using common algebraic techniques, expressions can be simplified. For example, consider the expression \(2x + 2\). It can be simplified by factoring out the common factor, resulting in \(2(x + 1)\). This step reduces the expression to its most straightforward form, making it easier to work with in subsequent calculations.

• Identify common factors to simplify the expression.
• Combine like terms, if possible, to reduce complexity.
• Simplification usually makes further calculations easier and more efficient.

Remember, the goal of simplification is to make algebraic expressions as concise and clear as possible, which often involves turning them into a product of simpler expressions.

Factoring in Algebra

Factoring is the process of breaking down an expression into a product of simpler expressions, called factors. It's like figuring out what numbers multiply together to create a larger number, but in the realm of algebra. In algebraic terms, to factor \(x^2 + x\), you should look for a common term in all parts of the expression. Here, you can factor out an \(x\), resulting in \(x(x + 1)\).

• Factoring is useful for solving equations, simplifying expressions, and finding zeros of functions.
• It involves finding the greatest common factor (GCF) of the terms.
• Check each term for common variables and constants that can be factored out.

Factoring not only simplifies the expressions but also sets the stage for further operations, such as division or solving equations.

Division of Algebraic Expressions

Division of algebraic expressions often involves changing the division operation into multiplication by the reciprocal. This method makes the calculation much more straightforward. For instance, in the expression \((2x+2) \div \frac{x^2+x}{4}\), the division can be rewritten as multiplication: \(2(x+1) \cdot \frac{4}{x(x+1)}\). This trick simplifies the problem by avoiding division directly and instead multiplying by the flipped fraction.

• Always change division into multiplication by flipping (reciprocating) the second fraction.
• Factor the expressions first to easily see what cancels out.
• Cancel out any common factors in the numerator and denominator.

This approach ensures each expression is dealt with in simpler steps, leading to a cleaner and more manageable final result.