105 LearnersLast updated on July 5th, 2025
Zero Product Property
The result of multiplying any number by zero is zero, and this property of multiplication is the zero product property. It can be represented as, if a × b = 0, then either a = 0 or b = 0 or a = b = 0; this is known as the zero product principle/rule.
What is Zero Product Property?
The zero product property is when the product of two or more factors is zero, then at least one of the factors is zero. The property applies to multiplication in equations, in matrices, and for vectors. It can be expressed as:
a × b = 0, then either a = 0 or b = 0 or both a = b = 0.
If (x + a) (x + b) (x + c) . . . (x + n) = 0, then one of the factors must be zero.
So, x + a = 0 or x + b = 0, or …, x + n = 0
What is Zero Product Property in Equations?
To solve algebraic equations, especially quadratic and polynomial equations, we use the zero product property. This property is used to find the values of the variables. To solve the quadratic equations in factored form, we use the zero product property. For example, if (x + a)(x + b) = 0, then according to zero product property, (x + a) = 0 or (x + b) = 0.
What is Zero Product Property in Matrices?
For real numbers, the product of two numbers is zero when at least one multiplier is zero. But it is not true for matrix multiplication. That is, the product of two matrices can be a zero matrix, but one of the matrices doesn't have to be a zero matrix. For example, let
Then AB is:
So, the product of A and B is the zero matrix, but neither A nor B is a zero matrix.
What is Zero Product Property in Vector?
Like matrices, the zero product property does not apply to vectors. This means if the dot product or cross product of two vectors is zero, at least one vector doesn't need to be a zero vector.
For example: let u = 2i + 3j and v is 3i - 2j
u v = (2)(3) + (3)(-2)
= 6 - 6 = 0
Here, the product is 0, but neither u nor v is non-zero.
What Are the Advantages and Disadvantages of Zero Product Property
Zero product property states that if the product of two or more factors is zero, then at least one of the factors is zero. Here are some of the advantages and disadvantages of the property.
|
Advantages |
Disadvantages |
|
Helps to solve algebraic equations by setting each factor to zero. |
Not applicable for matrices, which means that the product of two non-zero matrices can be 0. |
|
Used to simplify the quadratic and higher-degree polynomial equations. |
Even the zero product property does not apply to vectors; that is, the product of two non-zero vectors can be a zero vector. |
Real-world applications of Zero Product Property
The zero product property is a fundamental concept in algebra and is used in various fields. The real-world applications of the zero product property are:
- In physics, to find the projectile motion or kinematics, equations are often set equal to zero to find when an object hits the ground.
- In profit modeling, companies set the profit equation to zero to determine break-even points.
- To solve algebraic equations across STEM fields, we use the zero product property.
Common Mistakes and How to Avoid Them in Zero Product Property
It is common among students to make mistakes when applying the zero product property. Here are some common mistakes and ways to avoid them to master the zero product property.
Mistake 1
Forgetting to set each factor equal to zero
When factoring an expression, students sometimes forget to set each factor equal to zero. Zero product property states that either factor can be zero. That is, if AB = 0, then either A = 0 or B = 0. Always set each factor equal to zero when applying the property.
Mistake 2
Thinking zero product property applies to addition
Students sometimes think that zero property is applicable for both addition and multiplication, that is, a + b = 0 if a = 0 or b = 0. But it is not true, the zero product property is only applicable for multiplication, that is, a × b = 0, if a = 0 or b = 0.
Mistake 3
Thinking both factors must be zero
Students think that a × b = 0 only if a = 0 and b = 0, but it is not true. The zero product property states that if at least one multiplier is zero, then the product is zero. For example, if a = 5 and b = 0, then ab = 5 × 0 = 0.
Mistake 4
Confusing zero product property with zero exponent rules
Confusing zero product property and zero exponent rule is common among students, that is, students think a0 = 0 instead of 1. So students should remember that a × 0 = 0, which is the zero product rule, and a0 = 1 is the zero exponent rule.
Mistake 5
Assuming AB = 0 means A = 0 or B = 0 in matrices
One common error in matrix multiplication is assuming that if AB = 0, then A = 0 or B = 0, but this is not true for matrices. Matrices do not follow the zero product property like real numbers; that is, in matrices, if AB = 0, even if A and B are not zero matrices.
Solved Examples of Zero Product Property
Problem 1
The product of two numbers is zero. One of the numbers is (x – 3), and the other is (x + 2). What are the possible values of x?
x = 3 or x = -2
Explanation
Here, (x - 3)(x + 2) = 0, so either factor can be zero
x - 3 = 0 ⇒ x = 3
x + 2 = 0 ⇒ x = -2
Problem 2
The product of three expressions, (x + 1), (x + 4), and (x – 2) is zero. Find all possible values of x.
x = -1, x = -4, or x = 2
Explanation
As (x + 1)(x + 4)(x -2) = 0
Set each factor to zero, that is:
x + 1 = 0 ⇒ x = -1
x + 4 = 0 ⇒ x = -4
x - 2 = 0 ⇒ x = 2
Problem 3
Find the equation when the factors are (x - 2) and (x + 5).
x2 + 3x -10 = 0
Explanation
To find the equation, we multiply the factors,
(x - 2)(x + 5) = x2 + 5x - 2x - 10
= x2 + 3x - 10
Problem 4
Find the roots of the quadratic equation whose factored form is (x + 3)(x - 2) = 0.
x = -3 or x = 2
Explanation
Since (x +3)(x -2) = 0
That is x + 3 = 0 ⇒ x = -3
x - 2 = 0 ⇒ x = 2
Problem 5
If the factors are (x - 1) and (x + 6), find the equation
x2 + 5x - 6 = 0
Explanation
To find the equation, we multiply the factors.
(x - 1)(x + 6) = x2 + 6x - 1x - 6
= x2 + 5x - 6
FAQs on Zero Product Property
1.What is the zero product property?
2.Can I use the zero product property in addition?
3.What is the product of 154, 564, and 0?
4.Can we use the zero product property to solve a quadratic equation?
5.How is the zero product property used in real life?
6.How does learning Algebra help students in Vietnam make better decisions in daily life?
7.How can cultural or local activities in Vietnam support learning Algebra topics such as Zero Product Property?
8.How do technology and digital tools in Vietnam support learning Algebra and Zero Product Property?
9.Does learning Algebra support future career opportunities for students in Vietnam?
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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: He loves to play the quiz with kids through algebra to make kids love it.
