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Factor. \[ 125-27 w^{3} \]

Ask by Dunn Henry. in the United States
Mar 14,2025

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The expression \(125-27w^{3}\) factors to \((5-3w)(25+15w+9w^{2})\).

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To factor the expression \( 125 - 27w^3 \), we can recognize that this represents a difference of cubes. Remember that \( 125 = 5^3 \) and \( 27 = 3^3 \), so we can rewrite the expression as: \[ 5^3 - (3w)^3 \] Using the difference of cubes formula, which states \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \), we identify \( a = 5 \) and \( b = 3w \). Substituting these into the formula gives us: 1. The first factor is \( (a - b) = (5 - 3w) \). 2. The second factor is \( a^2 + ab + b^2 \): - \( a^2 = 5^2 = 25 \) - \( ab = 5(3w) = 15w \) - \( b^2 = (3w)^2 = 9w^2 \) Putting it all together, we have: \[ 125 - 27w^3 = (5 - 3w)(25 + 15w + 9w^2) \] So, the final factored form of \( 125 - 27w^3 \) is: \[ (5 - 3w)(25 + 15w + 9w^2) \]

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