What is 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.
More about Zero Product Property
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.
While we study mathematics by the use of symbols or variables for expressing different principles and formulas. When we equate an algebraic expression to zero then we solve the equation to find the value of the variable which will make the value of the expression zero. For this, we have to understand that the product of any number or expression is zero. Algebraic equations and algebraic expressions play a very major role in science and Maths. All the advanced concepts in these fields depend on the very basic concept of algebra. The whole branch of theoretical physics is based on solutions of algebraic equations only. Many important discoveries in science including many groundbreaking ones are done by simply solving algebraic equations. The famous theory of relativity is also found this way by solving an algebraic equation.
While solving any algebraic equation we use the method of breaking the expression into simple multiplication of zeros. Zeros is a simple algebraic equation where the value of the variable can be found easily by equating the expression with zero.
Definition of Zero Product Property
The zero product property definition in algebra states that the product of two nonzero elements is nonzero. In other words, this assertion:
If pq = 0, then either p or q = 0.
The zero product property is also known as the zero multiplication property, the null factor law, the nonexistence of nontrivial zero divisors, the zero product rule, or one of the two zero factor properties.
The zero product rule is satisfied by all number systems including rational numbers, integers, real numbers, and complex numbers. In general, a domain is a ring that satisfies the zero property.
How to make Use of the Zero Product Property?
According to the zero product rule, if the product of any number of expressions is 0, then at least one of them must also be zero. That is to say,
p.q.r = 0 denotes that p = 0 or q = 0 or r = 0
As a result, we solve quadratic equations by first setting them to 0.
This polynomial can now be factored into terms. These terms will have a product of zero, so we will use the zero product rule to find the roots of our equation.
Let's look at an example of how to use the zero product property:
To find the equation, use the zero product property:
y² + 5y = -4
To begin, set everything to zero as shown below:
y² + 5y + 4 equals 0
Then, factorise the left side as follows:
0 = (y + 4)(y + 1)
We know that at least one of the expressions (y + 4) and (y + 1) is equal to 0 because they multiply together to produce the result 0. This allows us to deduce the original equation.
y + 4 = 0→ y - 4 = 0
y + 1 = 0 → y - 1 = 0
As a result, the y (or roots) solutions are -4 and -1. We can test this by substituting it into the original equation:
y² + 5y = -4
Using y = - 4 as a check: (-4)
2+5(-4) = 16+20 = - 4
Checking for y = -1: (-1)2 + 5(-1) = 1 - 5 = -4
Examples of Zero Product Properties
The zero product property examples provided below will help you understand the zero product rule correctly.
1. Solve the equation 6y² + y -15 using the zero product property.
Solution: To begin, set everything to zero as shown below:
6y² + y - 15 equals 0
In order to solve variable y, factorise the left side:
(3y+5)(2y-3) = 0
We know that at least one of the expressions (3y + 5) and (2y -3) equals 0 because they are multiplied together to produce the result 0. This allows us to deduce the original equation.
3y + 5 = 0→ y = -5/3
2y - 3 = 0→ y = 3/2
As a result, the y (or roots) solutions are -5/3 and 3/2.
2. Solve the equation (y - 2)²(y -1)= 2(2y - 5) using the zero product property ( y - 2).
Solution: The preceding equation can be written as follows:
(y - 2)2(y -1) = 2(2y - 5)( y - 2)
y3 - 5y2 + 8y - 4 = 4y - 18y + 20
(y³ - 5y2 + 8y - 4) - (4y - 18y + 20y) = 0
y³ - 9y² + 26 y - 24 = 0
(y - 2) (y - 3) (y - 4) = 0
Using the rule of zero product, we get
(y - 2) = 0, (y - 3) = 0, and (y - 4) = 0.
As a result, the solutions (or roots) for y are y = 2 or y = 3 or y = 4.
FAQs on Zero Product Property
1. What is the Zero Product Property in Maths?
The Zero Product Property states that if the product of two or more factors is equal to zero, then at least one of those factors must be zero. In algebraic terms, if a × b = 0, it logically follows that either a = 0, or b = 0, or both a and b are equal to zero. This principle is fundamental for solving polynomial equations.
2. When should you apply the Zero Product Property while solving equations?
You should apply the Zero Product Property specifically when you have an equation where a polynomial is set equal to zero. The process involves first factoring the polynomial into a product of simpler expressions (like binomials). Once the equation is in the form (factor 1) × (factor 2) = 0, you can use the property to find the solutions by setting each factor to zero.
3. How is the Zero Product Property used to solve a quadratic equation?
To solve a quadratic equation like x² + 2x - 15 = 0 using this property, you would follow these steps:
- Step 1: Factor the equation. The quadratic expression x² + 2x - 15 factors into (x + 5)(x - 3). So, the equation becomes (x + 5)(x - 3) = 0.
- Step 2: Apply the Zero Product Property. Set each factor equal to zero individually. This creates two separate linear equations: x + 5 = 0 and x - 3 = 0.
- Step 3: Solve for the variable. Solving these two equations gives the roots of the quadratic equation: x = -5 and x = 3.
4. What is a real-world example of the Zero Product Property?
A common real-world example is in physics, when calculating the trajectory of a thrown object. The height of the object over time can be modelled by a quadratic equation. To find the times when the object is at ground level (height = 0), you set the equation to zero. Factoring the equation and using the Zero Product Property will give you two time values: the starting time (when it was launched from the ground) and the landing time (when it returns to the ground).
5. Why does the Zero Product Property only work when the product equals zero?
This property is unique to the number zero because of its special multiplicative identity: any number multiplied by zero results in zero. This certainty does not exist for any other number. For instance, if you know that a × b = 12, you cannot determine the values of a or b. They could be 3 and 4, 2 and 6, -1 and -12, or infinite other combinations. Only a product of zero guarantees that at least one of the contributing factors must be zero.
6. What is the most common mistake students make when using the Zero Product Property?
The most frequent error is applying the property to an equation that is not set equal to zero. For example, if given the equation (x - 4)(x + 2) = 7, it is incorrect to set x - 4 = 7 or x + 2 = 7. The correct method is to first expand the expression, subtract 7 from both sides to set the equation to zero (x² - 2x - 8 - 7 = 0), and then factor the new quadratic (x² - 2x - 15 = 0) before applying the property.
7. Does the Zero Product Property apply to all mathematical systems, such as matrices?
No, the Zero Product Property does not generally apply to matrix algebra. It is possible for two non-zero matrices to be multiplied together to result in a zero matrix (a matrix where all entries are zero). In this context, these non-zero matrices are known as 'divisors of zero'. This is a key distinction between the algebra of real numbers and matrix algebra.
