Applying the zero product property to solve equations

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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.

Does the zero product property apply for all equations?

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How do you explain the zero property?