Zero Product Property

Introduction to Zero Product Property

The zero product property, also known as the zero product principle, states that if p q = 0, then p = 0, q = 0, or both p and q = 0. When factoring in expressions from both sides, be cautious about cancelling the 0 (null) solutions.

The zero product property allows us to factor equations and solve them. For instance, x² - 6x + 5 = 0 or (x - 1) (x - 5) = 0.

With the zero product property, (x - 1) = 0 or (x - 5) = 0. As a result, the answers are x = 1 and x = 5.

However, the zero product property cannot be used in matrices because the product of two matrices P and Q can be 0.

Zero product property defines that if A and B are two real numbers and multiplication of A and B is zero then it must be either A=0 or B=0 and there might be some situations where A and B both are equal to zero. So, we can say that the multiplication of two non zero real numbers can never be zero. It is usually used in the solution of algebraic equations. An algebraic expression is any expression that involves any variable. An algebraic equation is an algebraic expression that can be equated to zero.  

Proposition 1.27 of Euclid's Elements, a theorem of absolute geometry (hence valid in both hyperbolic and Euclidean Geometry), proves that if the angles of a pair of alternate angles of a transversal are congruent then the two lines are parallel (non-intersecting).

It follows from Euclid's parallel postulate that if the two lines are parallel, then the angles of a pair of alternate angles of a transversal are congruent (Proposition 1.29 of Euclid's Elements).

What is Zero Product Property

The Zero Product Property states that if the product of two numbers is zero, then at least one of the numbers is zero. In symbols, where a and b represent numbers, if ab=0, then a = 0 or b=0.

When the lines are parallel, a case that is often considered, a transversal produces several congruent and several supplementary angles. Some of these angle pairs have specific names and are discussed below.

Corresponding angles, alternate angles, and consecutive angles.

If the two angles of one pair are congruent (equal in measure), then the angles of each of the other pairs are also congruent.  

The zero-product property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties. All of the number systems studied in elementary mathematics, the integers Z, the rational numbers Q, the real numbers R, and the complex numbers C. This satisfies the zero-product property. In general, a ring which satisfies the zero-product property is called a domain.

Euclid's Proposition 27 states that if a transversal intersects two lines so that alternate interior angles are congruent, then the lines are parallel. Euclid proves this by contradiction: If the lines are not parallel then they must intersect and a triangle is formed.

Examples of zero product properties

The zero-product property, also called zero-product principle, states that for any real numbers a and b, if ab = 0, then either an equal zero, b equals zero, or both a and b equal zero. 

Thus, if we have a linear equation in slope-intercept form, it's easy to see that the slope of the line is the constant in front of the x variable. For example, suppose that price of a product over x years is modeled by the equation y = 0.2x + 5, where y is the price of the product, and x is the number of years that have passed since the company started selling that product.

Deriving the Standard Form for a Linear Equation

The standard form of a line can be particularly helpful when solving a system of equations. For instance, when using the elimination method to solve a system of equations, we can easily align the variables using standard form.

In the case of two variables, each solution may be interpreted as the Cartesian coordinates of a point of the Euclidean plane. The solutions of a linear equation form a line in the Euclidean plane, and, conversely, every line can be viewed as the set of all solutions of a linear equation in two variables. This is the origin of the term linear for describing this type of equations. More generally, the solutions of a linear equation in n variables form a hyperplane (a subspace of dimension n − 1) in the Euclidean space of dimension n.

This article considers the case of a single equation with coefficients from the field of real numbers, for which one studies the real solutions. All of its content applies to complex solutions and, more generally, for linear equations with coefficients and solutions in any field.

Proposition 1.27 of Euclid's Elements, proves that if the angles of a pair of alternate angles of a transversal are congruent then the two lines are parallel (non-intersecting).

The functions whose graph is a line are generally called zero product property in the context of calculus. However, in linear algebra, a linear function is a function that maps a sum to the sum of the images of the summands. So, for this definition, the above function is linear only when c = 0, that is when the line passes through the origin. For avoiding confusion, the functions whose graph is an arbitrary line are often called affine functions.

With this interpretation, all solutions of the equation form a line, provided that a as well as b are not both zero. Conversely, every line is the set of all solutions of a linear equation.

The phrase "zero product property" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line.