# Zero Product Property

## What is Zero Product Property?

Zero product property, also known as zero product principle states that if p × q = 0 , then p = 0 or q = 0 or both p = 0 and q = 0 .

When factoring in expressions from both sides, one should be careful while cancelling the 0 ( null) solutions.

Zero product property helps us to solve equations through factoring. For example, we have x² - 6x + 5 = 0 or (x - 1) (x - 5) = 0.

Using the zero product property, either (x - 1) = 0 or (x - 5) = 0. Therefore, the solutions are x = 1 and x = 5.

However, zero product property may not be applied in matrices because two matrices P and Q can have a product of 0.

### Zero Product Property Definition

In algebra, zero product property definition states that the product of two non zero elements is nonzero. In other words, this statement that:

If pq = 0, then p = 0 or q = 0

The zero product property is also known as the multiplication property of zero, the null factor law, the nonexistence of non-trivial zero divisors, the rule of zero product, or one of the two zero factor properties.

All the number systems such as rational numbers, integers, real numbers, complex numbers satisfy the zero product rule. Generally, the ring that satisfies zero property is known as a domain.

### How to Use Zero Product Property?

The zero product rule states that if the product of any number of expressions is 0, then at least one of them must be zero. In other words,

p.q.r = 0 means p = 0 or q = 0 or r = 0

Due to this, we solve quadratic equations by first setting them equal to 0.

Now, we can factor this polynomial into terms. These terms will have a product of 0, Hence, we will find the roots of our equation using zero product rule.

Let us understand how to use zero product property through an example:

Use zero product property to find the equation:

y2 + 5y = - 4

First, set everything equal to zero as shown below:

y2 + 5y + 4 = 0

Then, factorize the left hand side:

(y + 4)( y + 1) = 0

As the expression (y + 4) and (y + 1) multiply together to obtain the result 0, we know at least one of them equal to 0. This enables us to find out the original equation.

y + 4 = 0 → y = - 4

y + 1 = 0 → y = - 1

Therefore, the solutions for y (or roots) are - 4 and -1. We can substitute this in the original equation and verify:

y2 + 5y = - 4

Checking y = - 4: (-4)2 + 5(-4) = 16 + 20 = - 4

Checking y = -1: (-1)2 + 5(-1) = 1 - 5 = - 4.

### Zero Product Property Examples

The zero product property examples given below enable you to understand the zero product rule appropriately.

1. Use the zero product property to solve the equation 6y2 + y -15.

Solution:

First, set everything equal to zero as shown below:

6y2 + y - 15 = 0

Further, factorize the left-hand side to solve variable y:

(3y + 5)(2y -3) = 0

As the expression (3y + 5) and ( 2y -3) multiplied together to obtain the result 0, we know at least one of them equal to 0. This enables us to find out the original equation.

3y + 5 = 0 → y = -5/3

2y - 3 = 0 → y = 3/2

Therefore, the solutions for y (or roots) are -5/3 and 3/2.

2. Use the zero product property to solve the equation (y - 2)2(y -1)= 2(2y - 5)( y - 2).

Solution:

The above equation can be written as:

(y - 2)2(y -1) = 2(2y - 5)( y - 2)

y3 - 5y2 + 8y - 4 = 4y - 18y + 20

(y3 - 5y2 + 8y - 4) - (4y - 18y + 20) = 0

y3 - 9y2 + 26 y - 24 = 0

(y - 2) (y - 3) (y - 4) = 0

Using the zero product rule, we get

(y - 2) = 0 or (y - 3)= 0 or (y - 4) = 0

Therefore, the solutions for y (or roots) are y = 2 or y = 3 or y = 4.

Q1. What Does the Principle of Zero Product Property States?

Ans. The principle of zero product property states that if there is the product of any two numbers, that is equivalent to 0, then either the first number, or the second number, or both has to be 0.

The principle of zero product is useful to solve any equation.

Example: (x + 5)( x - 3)(x - 2) = 0, then x = -5, x = 3 , and x = 2

The property is valid for all the number systems in Mathematics.

Q2. How to Solve Variables in Quadratic Equations Using Zero Product Property?

Ans. When solving variables in quadratic equations, rewrite the given equation in factored form and set them equal to zero. Using the zero product property, if one factor is equal to zero, then the product of all the factors is equal to zero.

Moving further, when you have set all the binomial factor equal to 0 and have solved the variables in the equation, then all the possible solutions of the equations have been found. Some solutions may not be viable, so be sure to find if each solution is suitable for the problem.

Q3. When do we Use Zero Product Property?

Ans. We use zero product property to solve quadratic equations. It is because factoring the equation gives us two expressions that multiply together to be 0. We can solve the variable 'x' by setting each factor equal to 0.