The Zero Product Property states that if the product of two or more factors equals zero, then at least one of the factors must be equal to zero. In other words, if \( a \times b = 0 \), then either \( a = 0 \), \( b = 0 \), or both.
To apply the Zero Product Property, follow these steps:
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Set the Equation to Zero: Make sure your equation is in the form \( f(x) = 0 \), where \( f(x) \) is a polynomial.
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Factor the Polynomial: Factor the polynomial into two or more factors.
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Set Each Factor to Zero: Once you have factored the equation, set each factor equal to zero.
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Solve for the Variable: Solve each equation to find the possible values of the variable that make the original equation true.
Quick Example:
Consider the equation: \[ x^2 - 5x + 6 = 0 \]
Step 1: Factor the quadratic: \[ (x - 2)(x - 3) = 0 \]
Step 2: Set each factor to zero: \[ x - 2 = 0 \quad \text{or} \quad x - 3 = 0 \]
Step 3: Solve each equation: \[ x = 2 \quad \text{or} \quad x = 3 \]
Solution: The solutions are \( x = 2 \) and \( x = 3 \).
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