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Zero Product Property
Zero product property is applied to find the individual factor value. If an expression satisfies the zero product property then it is equal to zero, and it has solutions. Satisfying this property signifies that on one side of the equals to symbol we have an expression that is a product of factors and on the other side it is equal to zero.
Zero product property is applicable to algebraic equations but not to matrices or vectors. Let us check more about it through examples, FAQs.
What is Zero Product Property?
Zero product property has one side of the expression equal to zero and the other side is the product of two or more factors. This property applies to multiplication in algebra, in matrices, and for vectors. The zero product property says that if the product of two or more factors is equal to zero then at least one of the factors is equal to 0 (because otherwise, the product won't be equal to 0). i.e.,
The zero product property can be further extended to more factors and it looks like below in that case.
Whenever (x + a)(x + b)(x + c)....(x + n) = 0 ⇒ x + a = 0 (or) x + b = 0 (or) .... (x + n) = 0
Note that, more than one of the factors may also be equal to zero for the product to be 0. The application of zero product property can be done for equations, but cannot be applied to matrices and vectors.
Zero Product Property in Equations
Zero product property for equations is helpful to solve the equation and find the values of the variables. The algebraic expression following the zero product property has factors and can also be solved to find the values of the variables. Zero product property is very helpful in solving the quadratic equations that are in the factored form. For example, if (x + p) (x + q) = 0, then by zero product property, we can say that x + p = 0 or x + q = 0 and solving each of these for x would give the solutions of the given quadratic equation.
Similarly, the zero product property can be applied to polynomial equations. For example, if x (x + 1) (x + 2) = 0 then by the application of zero product property, x = 0 (or) x + 1 = 0 (or) x + 2 = 0 which gives x = 0, 1, and 2 as roots.
Zero Product Property in Matrices
The zero product property is not applicable for matrices. i.e., though the product of two matrices is a zero matrix, it is not compulsory that one of the matrices should be a zero matrix. i.e., the zero product property cannot be applied for the multiplication of matrices. Consider the following example.
Let A = \(\left [ {\begin {array} {cc} 0 & 1 \\ \\ 0 & 0 \\ \end {array} } \right]\) and B = \(\left [ {\begin {array} {cc} 0 & 0 \\ \\ 0 & 1 \\ \end {array} } \right] \). We can verify that AB = O.
AB = \(\left [ {\begin {array} {cc} 0 & 1 \\ \\ 0 & 0 \\ \end {array} } \right] \) \(\left [ {\begin {array} {cc} 0 & 0 \\ \\ 0 & 1 \\ \end {array} } \right] \) = \(\left [ {\begin {array} {cc} 0 & 0 \\ \\ 0 & 0 \\ \end {array} } \right] \)
But observe that none of A and B is actually a zero matrix.
Zero Product Property in Vectors
The zero product property cannot be applied for vectors as well. Whenever the dot product or cross product of any two vectors is 0, it doesn't mean that at least one of the vectors is a zero vector. Consider the following examples.
Example 1: For a = i + j and b = i  j, a · b = (i + j) · (i  j) = 1  1 = 0, but neither a nor b is a zero vector.
Example 2: We know that i × i = 0, but neither of the vectors is a zero vector in this case. In fact, i is a unit vector.
Merits and Demerits Of Zero Product Property
The following are some of the important merits and demerits of zero product property.
 Zero product property is applicable to find the values of the variables in an algebraic equation by setting each of the factors to 0.
 But to solve an equation using the zero product property, one must be aware of the process of factorizing the expressions.
 The zero product property cannot be applied to matrices or vectors.
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Examples of Zero Product Property

Example 1: Solve the quadratic equation 2x^{2}  5x  3 = 0 by zero product property.
Solution:
We will factorize the quadratic expression in the given equation. Then we get
(2x + 1) (x  3) = 0
By zero product property,
2x + 1 = 0 or x  3 = 0
x = 1/2 or x = 3
Answer: 1/2 and 3 are the roots. 
Example 2: Find the equation whose factors are (x + 4), and (3x  5).
Solution:
The given two factors are (x + 4), and (3x  5).
By the zero product property, the product of these two expressions is equal to zero.
(x + 4)(3x  5) = 0
x(3x  5) + 4(3x  5) = 0
3x^{2}  5x + 12x 20 = 0
3x^{2} + 7x  20 = 0
Answer: Therefore the equation is 3x^{2} + 7x  20 = 0

Example 3: Find the roots of the cubic equation which is in factored form is (x + 1)^{2} (2x  3) = 0.
Solution:
We can write the given equation as (x + 1) (x + 1) (2x  3) = 0.
By the zero product property, x + 1 = 0, x + 1 = 0, 2x  3 = 0.
This gives x = 1, x = 1, x = 3/2
Answer: The roots of the given cubic equation are 1, 1, and 3/2.
FAQs on Zero Product Property
What is Zero Product Property in Algebra?
The zero product property in algebra is applicable across quadratic equations and polynomial equations. This property says for any two expressions 'a' and 'b', whenever a × b = 0, either a = 0 or b = 0. This property is useful in solving the quadratic equations, cubic equations, etc after factoring.
How to Apply Zero Product Property?
The zero product property can be applied when the product of the expressions is equal to zero. This property helps in equalizing the individual factors of the expression to zero and then solving it.
Can Zero Product Property be Applied Anywhere in Math?
No, the zero product property is applicable only to solve the algebraic equations. But it is applicable to neither matrices nor to vectors.
What Does Zero Product Property State?
The zero product property states that if there is the product of factors on one side and 0 on the other side of an equation, then at least one of the factors must be equal to 0. This statement is applicable across higher degree equations as well but not to matrices and vectors.
What is The Use Of Zero Product Property?
The zero product property is useful to find the roots of a polynomial equation. But to apply this property, we need to factorize the left side part of the polynomial equation and make the right side part to be 0.
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