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Zero product principle

If the product of two expressions is zero, then at least one of the expressions must be zero.

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Student-friendly explanation

This principle is the bridge between factorised form and final roots. Once a quadratic equation is written as a product, each factor is set to zero separately.

How to write this in exams

  1. 1

    Start with the exact idea

    If the product of two expressions is zero, then at least one of the expressions must be zero.

  2. 2

    Then show how to use it

    1. Write the equation in product form. 2. Use the zero product principle. 3. Set each factor equal to zero. 4. Solve each linear equation. 5. Write all roots clearly.

  3. 3

    Add one concrete example

    (x - 4)(x + 2) = 0 gives x = 4 or x = -2.

  4. 4

    Avoid this incomplete answer

    Solving (x - 7)(x + 2) = 0 by saying x - 7 = x + 2 = 0 is not a proper method.

Definition

If the product of two expressions is zero, then at least one of the expressions must be zero.

Example

(x - 4)(x + 2) = 0 gives x = 4 or x = -2.

Rule to remember

A × B = 0 implies A = 0 or B = 0. This is the zero product principle used after factorisation.

Memory hook

Zero product means zero in one bracket at least.

Examples and method

Worked example

For (x - 7)(x + 2) = 0, set x - 7 = 0 and x + 2 = 0. Then x = 7 and x = -2. These are the solutions because each factor separately makes the product zero.

Method to apply

1. Write the equation in product form. 2. Use the zero product principle. 3. Set each factor equal to zero. 4. Solve each linear equation. 5. Write all roots clearly.

Diagram support

A simple branching arrow from one product into two equations helps students remember the split.

How CBSE asks it

The principle is often hidden inside factorisation questions and is also used in short-answer reasoning steps.

Avoid common mistakes

Common confusion

Students sometimes solve by adding the factors instead of setting each factor to zero.

Common wrong answer

Solving (x - 7)(x + 2) = 0 by saying x - 7 = x + 2 = 0 is not a proper method.

Exam tip

After factorisation, never keep the product equal to zero only; split it into two separate equations.

Quick check

What should you do when (x - 7)(x + 2) = 0 appears in a question?

Set each factor equal to zero separately. Then solve x - 7 = 0 and x + 2 = 0, because a product becomes zero only when at least one factor is zero.

Answer writing and exam use

1-mark answer

If the product of two expressions is zero, then at least one of the expressions must be zero.

2-mark answer

If the product of two expressions is zero, then at least one of the expressions must be zero. A × B = 0 implies A = 0 or B = 0. This is the zero product principle used after factorisation. (x - 4)(x + 2) = 0 gives x = 4 or x = -2.

3-mark answer

This principle is the bridge between factorised form and final roots. Once a quadratic equation is written as a product, each factor is set to zero separately. A × B = 0 implies A = 0 or B = 0. This is the zero product principle used after factorisation. For (x - 7)(x + 2) = 0, set x - 7 = 0 and x + 2 = 0. Then x = 7 and x = -2. These are the solutions because each factor separately makes the product zero. The principle is often hidden inside factorisation questions and is also used in short-answer reasoning steps. Solving (x - 7)(x + 2) = 0 by saying x - 7 = x + 2 = 0 is not a proper method.
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