Problem 30 - Algebra and Trigonometry

Question

Find the sum, difference, or product. \(-y\left(y^{2}-2\right)\)

Step-by-Step Solution

Answer

The expression simplifies to \(-y^3 + 2y\).

Step 1: Identify the Expression

The expression given is \(-y(y^2 - 2)\). It involves the multiplication of \(-y\) with the binomial \(y^2 - 2\). Our task is to perform this multiplication.

Step 2: Distribute -y into the Binomial

Use the distributive property to multiply each term inside the parenthesis by \(-y\).This means multiplying \(-y\) by \(y^2\) and by \(-2\).

Step 3: Multiply -y by y^2

Multiply \(-y\) and \(y^2\):\(-y \cdot y^2 = -y^{1+2} = -y^3\).This is because, when multiplying powers with the same base, you add the exponents.

Step 4: Multiply -y by -2

Multiply \(-y\) and \(-2\):\(-y \cdot -2 = 2y\).The product of two negative numbers is a positive number.

Step 5: Write the Simplified Expression

Combine the results from the previous steps to write the complete expression:\(-y^3 + 2y\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that simplifies expressions and helps us perform mathematical operations efficiently. Let's break down the steps to understand how it works. In algebraic terms, the distributive property allows you to multiply a sum or difference by multiplying each term separately and then adding or subtracting the results. It's represented by:

• When you have an expression like \(a(b+c)\), it becomes \(ab + ac\).
• Similarly, \(a(b-c)\) becomes \(ab - ac\).

In our exercise, we apply the distributive property to \(-y(y^2 - 2)\). By multiplying \(-y\) with each term inside the parentheses, we distribute the multiplication across both \(y^2\) and \(-2\). This action transforms the original expression into two simpler parts, \(-y \cdot y^2\) and \(-y \cdot (-2)\). As you practice using the distributive property, it becomes a powerful tool to tackle more complex problems smoothly.

Binomial

A binomial is a specific type of polynomial that contains exactly two terms separated by a plus or minus sign. Understanding binomials is essential in algebra since many operations, such as addition, subtraction, and multiplication, extend from manipulating these two-term expressions.In the given exercise, our binomial is \(y^2 - 2\). Let's break it down:

• The first term is \(y^2\), which is a variable raised to the second power.
• The second term is a constant, \(-2\).

When working with binomials, you often use the distributive property to expand expressions or solve problems involving multiplication. In this task, it's crucial to properly distribute \(-y\) over every part of our binomial to ensure that each term is accurately multiplied. This careful attention allows us to approach more intricate algebraic expressions with confidence and precision.

Exponent Rules

Exponent rules, or laws of exponents, are guidelines that help simplify expressions involving powers. These rules are especially crucial when multiplying polynomials, as seen in this exercise.Let's explore some vital exponent rules applied here:

• Multiplying like bases: When you multiply like bases, you add their exponents. That's why, in our exercise, \(-y \cdot y^2\) becomes \(-y^3\). Here, we're using the rule that \(a^m \cdot a^n = a^{m+n}\).
• Negative times a negative: While not directly an exponent rule, it's essential to note that multiplying two negative numbers results in a positive. This principle helped us solve \(-y \cdot (-2)\), which became \(+2y\).

Remembering and applying exponent rules helps in managing expressions with multiple terms and achieving correct simplifications. These rules become indispensable tools when dealing with polynomial operations and will support you in solving a wide range of algebraic problems with ease.