ZERO-PRODUCT PROPERTY

Definition of Zero-Product Property

If the product of two or more factors is zero then at least one of the factors should be zero. This property is known as Zero-Product Property.

That is, if XY = 0, then X = 0 or Y = 0 or both X and Y are equal to 0.
Similarly, if XYZ = 0, then X = 0 or Y = 0 or Z = 0 or all three are 0.
This thought can be extended to any number of factors.

More About Zero-Product Property

    The x-coordinate of the point where a line intersects the y-axis is 0. So, the y-intercept of a line can also be found by substituting x = 0 in the equation of the line.

Video Examples: The Zero-Product Property


Examples of Zero-Product Property

Ques: Consider this simple algebra problem. Solve x2 – 4x = 0.

Solution:

    Let’s solve this equation through factoring, followed by the application of the Zero-Product Property
    x2– 4x = 0 ⇒ x (4 – x) = 0 [Taking out the common variable x]
    ⇒ x = 0 or (4 – x) = 0 [Recall the definition of Zero-product Property.]
    ⇒ x = 0 or x = 4
    So, the solutions are x = 0 or x = 4
    Both these values satisfy the original equation x2 – 4x = 0

Solved Example on Zero-Product Property

Ques: Solve for x: x2 + 4x – 5 = 0

Solution:

    x2 + 4x – 5 = 0
    ⇒ x2 + 5x – x – 5 = 0 [Splitting the middle term]
    ⇒ x(x + 5) – 1(x+5) = 0 [Taking out the common factors]
    (x + 5)(x – 1) = 0 [Factor.]
    ⇒ x + 5 = 0 or (x – 1) = 0 [Recall the definition of Zero-product property.]
    ⇒ x = –5 or x = 1
    So, the solutions are x = –5 or x = 1
    Both these values satisfy the original equation x2 + 4x – 5 = 0.