Problem 48 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ -y\left(-y^{2}+y\right) $$

Step-by-Step Solution

Answer

The expression \( -y(-y^{2}+y) \) simplifies to \( y^{3}-y^{2} \) after applying the distributive property.

Step 1: Identify the terms to distribute

In the given expression \( -y(-y^{2}+y) \), \( -y \) is distributed to each term inside the parentheses \( -y^{2} \) and \( y \).

Step 2: Apply the distributive property

Next, distribute \( -y \) across the terms within the parentheses by multiplying \( -y \) to each of the two terms separately. This results in \( y^{3}-y^{2} \).

Step 3: Write the final answer

The final simplified expression after applying the distributive property is \( y^{3}-y^{2} \).

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in algebra that involves combining like terms and using arithmetic operations to make expressions more manageable. This process often involves the distributive property, which enables us to multiply a single term by each of the terms within parentheses. This is pivotal when dealing with complex expressions.

For example, consider the expression \( -y(-y^2+y) \). To simplify, one would first use the distributive property to eliminate the parentheses by multiplying \( -y \) by each term inside the parentheses resulting in \( y^3-y^2 \). Remember, to combine like terms which are terms that have the same variable raised to the same power, requires them to be present. In our case, after distribution, no further simplification is required as the resulting terms are not like terms.

Multiplying Polynomials

When multiplying polynomials, it is essential to understand that we are working with expressions consisting of multiple terms. A 'term' refers to a product of numbers and variables like \( -y^2 \) or \( y \). In the expression \( -y(-y^2+y) \), the initial term \( -y \) is distributed to both terms inside the parentheses, following the principle of the distributive property. Each term must be multiplied individually, and the process is similar to distributive multiplication.

Visualizing the Process

Imagine 'handshakes' between the term outside the parentheses and each term inside—one handshake per pair. This visualization helps students to keep track of their steps as they multiply individual term pairs. The challenge lies in keeping track of signs and applying the correct operation, particularly when dealing with negative numbers and variable exponents.

Properties of Operations

Understanding properties of operations, such as the distributive property, is crucial in algebra. The distributive property allows us to multiply a sum or difference by a number, distributing the number to each term within the parentheses. This property provides a framework to deal with complex equations systematically. There are other properties of operations, such as commutative, associative, and identity properties, but the distributive property is especially useful for simplifying expressions and multiplying polynomials.

Real-life Applications

In real-life scenarios, the distributive property can be used for quick mental calculations. For instance, when figuring out the cost of several items with the same price, instead of adding the price multiple times, one can multiply the price by the number of items. Clear comprehension of these properties is the foundation that makes algebra more approachable and solves problems more efficiently.