Problem 7 - Beginning and Intermediate Algebra

Question

Fill in each blank with one of the following. positive,negative,0 If three positive numbers, five negative numbers, and zero are multiplied, the product is ______

Step-by-Step Solution

Answer

0

Step 1: Identify the Numbers

Determine the types of numbers involved. The problem states that there are three positive numbers, five negative numbers, and zero.

Step 2: Understand the Effect of Zero

Recall that multiplying any number by zero results in zero. This is true regardless of the other numbers involved.

Step 3: Calculate the Product

Multiply the numbers together. Since zero is one of the factors, the entire product will be zero.

Key Concepts

Positive Numbers

Positive numbers are numbers greater than zero. Simply put, they are the numbers on the right side of a number line. They represent quantities, counts, and measurements that increase, such as scoring points in a game or earning money.

Here are a few key points about positive numbers:

• They are written without any sign or with a '+' sign.
• Examples include 1, 2, 3, and so on.
• Multiplication of two positive numbers always results in another positive number. For example, 3 * 4 = 12.
• When you add positive numbers, the result is always positive.

Understanding positive numbers, in this exercise, realizes that the three numbers mentioned as positive contribute their value positively to any calculation unless zero is involved.

Negative Numbers

Negative numbers are numbers less than zero. They are located on the left side of the number line and often represent a deficit or a loss, such as owing money or temperatures below freezing.

Key characteristics of negative numbers include:

• They are preceded by a '-' sign to distinguish them from positive numbers.
• Examples include -1, -2, -3, and so forth.
• The multiplication of two negative numbers always results in a positive number. For example, (-2) * (-3) = 6.
• The multiplication of a negative number by a positive number results in a negative number. For example, (-2) * 4 = -8.

In the provided exercise, there are five negative numbers. Without considering zero, these would alter the sign of the final number depending on how many negative numbers are present.

Multiplication Properties

Multiplication has several important properties that help simplify calculations and understand outcomes.

Here are a few key properties that are essential in multiplication:

• Commutative Property: It states that changing the order of the numbers doesn't change the product. For instance, 2 * 3 = 3 * 2.
• Associative Property: It suggests that the way numbers are grouped doesn't affect the product. For example, (2 * 3) * 4 = 2 * (3 * 4).
• Distributive Property: This property allows for breaking down a multiplication operation into smaller parts. For example, 2 * (3 + 4) = (2 * 3) + (2 * 4).
• Multiplicative Identity Property: According to this property, any number multiplied by 1 remains unchanged. For instance, 5 * 1 = 5.
• Multiplication by Zero: The key property relevant to our exercise is that any number multiplied by zero results in zero. This overrides all other numbers involved in the multiplication.

In our exercise, this property means that the inclusion of zero in the multiplication of three positive numbers and five negative numbers ensures the product is zero. Zero effectively 'nullifies' the product regardless of the presence of positive and negative numbers.