The Zero Product Rule is a fundamental principle in algebra, specifically used when solving quadratic equations. Here’s a detailed, step-by-step explanation to understand the statement of the product rule:
1. Understanding the Zero Product Rule:
The Zero Product Rule states that if the product of two numbers is zero, then at least one of the numbers must be zero. This rule is essential for solving polynomial equations where the polynomial can be factored into a product of linear factors.
2. Given Statements:
Let's analyze each option provided:
- Option A: If [tex]\( a b = 0 \)[/tex], then either [tex]\( a = 0 \)[/tex] or [tex]\( b = 0 \)[/tex], but not both.
- This statement is incorrect as it excludes the possibility of both [tex]\( a \)[/tex] and [tex]\( b \)[/tex] being zero simultaneously.
- Option B: If [tex]\( a b = 0 \)[/tex], then [tex]\( a = 0 \)[/tex].
- This statement is incorrect because it disregards the possibility that [tex]\( b \)[/tex] could be zero.
- Option C: If [tex]\( a b = 0 \)[/tex], then either [tex]\( a = 0 \)[/tex] or [tex]\( b = 0 \)[/tex], or both.
- This statement correctly captures the essence of the Zero Product Rule. It acknowledges all the scenarios when the product could be zero: [tex]\( a \)[/tex] could be zero, [tex]\( b \)[/tex] could be zero, or both [tex]\( a \)[/tex] and [tex]\( b \)[/tex] could be zero.
- Option D: If [tex]\( a b = C \)[/tex], then [tex]\( a = 0 \)[/tex] and [tex]\( b = 0 \)[/tex].
- This statement is incorrect because it changes the equation to [tex]\( a b = C \)[/tex], where [tex]\( C \)[/tex] could be any non-zero constant, so neither [tex]\( a \)[/tex] nor [tex]\( b \)[/tex] needs to be zero for the product to be equal to a constant [tex]\( C \)[/tex].
3. Conclusion:
After carefully analyzing all the provided options, we can determine that the correct statement of the product rule is:
Option C: If [tex]\( a b = 0 \)[/tex], then either [tex]\( a = 0 \)[/tex] or [tex]\( b = 0 \)[/tex], or both.
Therefore, Option C is the correct statement of the Zero Product Rule.
