Problem 15 - Algebra A Combined Function

Question

Multiply. (2)(-1)(-3)(0)

Step-by-Step Solution

Answer

The product is 0 because multiplying by zero gives zero.

Step 1: Identify the Effect of Multiplying by Zero

In any multiplication, if one of the numbers being multiplied is zero, the entire product will be zero. So, as soon as we notice the presence of zero in the sequence (2)(-1)(-3)(0), we can conclude the result without further calculations.

Step 2: Understanding the Zero Product Property

The property we used in the previous step is known as the Zero Product Property. Now, let's reaffirm this property: When you multiply any number by zero, the result is always zero. This confirms that any multiplication involving zero results in a product of zero.

Step 3: Confirm Other Operations Are Unnecessary

Since multiplying by zero leads to a result of zero, we don't need to calculate the product of the other numbers (2, -1, and -3). The presence of any zero already determines the overall product.

Key Concepts

Multiplication

Multiplication is one of the four basic arithmetic operations, allowing us to find the total number of objects in equal groups. It can be thought of as repeated addition. For instance, when we multiply 2 by 3, we are essentially adding the number 2, three times:

• 2 + 2 + 2

This equals 6.

When working with negative numbers in multiplication, remember these rules:

• A positive number times a positive number is positive.
• A negative number times a negative number is positive.
• A positive number times a negative number is negative.

These rules are crucial in understanding the outcome of multiplying different types of numbers. In the example given (2)(-1)(-3)(0), recognizing these rules helps us see how multiplication would typically proceed if not for the presence of zero.

Zero Property

The Zero Property of Multiplication, also known as the Zero Product Property, states that the product of any number and zero is always zero. This property is pivotal in mathematics because it simplifies many calculations.

In our example, (2)(-1)(-3)(0), as soon as zero is identified as one of the multipliers, the entire product becomes zero, regardless of the other numbers in the operation.
Understanding this property saves time and effort, as it allows you to determine the result of the multiplication quickly once zero is encountered. Remember, the presence of zero in any multiplication immediately dictates the outcome as zero, making further calculations redundant.

Algebra Concepts

Algebra introduces a more generalized way to work with numbers and multiplication. Instead of working with fixed numbers, algebra uses variables to represent numbers in equations and expressions. One of the fundamental ideas in algebra is how properties like the Zero Product Property help solve equations.

For example, if you come across an equation like

• \(x imes y = 0\)

you can apply the Zero Product Property. This lets you conclude that either

• \(x = 0\)
• or
• \(y = 0\)

Understanding how the Zero Product Property applies in algebraic settings is key to solving many equations and simplifying complex problems. You often do not need to find each component individually if you recognize the fast track provided by this property whenever zero is involved.