Zero-Product Property

The zero product law states that if the product of two numbers or algebraic expressions equals zero, then at least one of the factors must be zero. $$ a \cdot b = 0 \quad \Rightarrow \quad a = 0 \, \text{or} \, b = 0 $$

For instance, if we multiply \(3 \cdot 0\), the result is 0, and the same happens with \(0 \cdot 5\).

$$ 3 \cdot 0 = 0 $$

$$ 5 \cdot 0 = 0 \\ \vdots $$

It’s impossible to get a product of zero unless at least one of the numbers (or both) is zero.

This is because zero is the absorbing element in multiplication. The product of any number $n$ and zero $0$ will always be zero.

$$ n \cdot 0 = 0 \cdot n = 0 $$

Though it seems simple, this fundamental property of algebra has wide-reaching implications, from solving polynomial equations to applications in geometry.

Why does this matter? This concept comes in handy for solving a variety of problems in algebra and mathematical analysis. You can use it to solve equations and systems of equations or to prove the properties of functions and other mathematical structures.

Here’s a practical example of how the zero product law works.

Suppose you need to solve the equation

$$ (x-2) \cdot (x+3) = 0 $$

According to the zero product law, the product equals zero if at least one of the factors is zero.

So, if (x-2)(x+3)=0, it means either (x-2)=0 or (x+3)=0.

Now, you can solve each equation to determine when each factor becomes zero.

The first equation is zero when x=2

$$ x -2 = 0 $$

$$ x - 2 + 2 = 0 + 2 $$

$$ x = 2 $$

The second equation is zero when x=-3

$$ x + 3 = 0 $$

$$ x + 3 - 3 = 0 - 3 $$

$$ x = -3 $$

Therefore, the product (x-2)(x+3) equals zero when x=2 or x=-3

These are the values of x that make one of the factors zero, which in turn makes the entire product zero.

The zero product law is especially useful for solving algebraic equations, particularly polynomial equations.

Let’s look at another example.

Consider the classic quadratic equation:

$$ x^2 - 5x = 0 $$

You can factor the equation by pulling out the common factor $x$.

$$ x(x - 5) = 0 $$

Now, you can solve each equation separately to find which values make them zero:

The first equation $x=0$ is already solved, while the second equation $x-5$ becomes zero when $x=5$.

So, the solutions to the quadratic equation are \(x = 0\) and \(x = 5\).

This simple process allows you to solve a wide range of equations by factoring polynomials into simpler expressions.