To find the linear factors of the polynomial \(2x^3 - 24x^2 + 64x\), let's first factor it.
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Factor out the greatest common factor (GCF): The GCF of the terms is \(2x\): \[ 2x(x^2 - 12x + 32) \]
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Now factor the quadratic \(x^2 - 12x + 32\): We need to find two numbers that multiply to \(32\) and add to \(-12\). These numbers are \(-8\) and \(-4\): \[ x^2 - 12x + 32 = (x - 8)(x - 4) \]
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Putting it all together: Thus, the complete factorization of the polynomial is: \[ 2x(x - 8)(x - 4) \]
From the options provided, the correct factorization of the polynomial \(2x^3 - 24x^2 + 64x\) in linear factors is: \[ \boxed{2x(x−8)(x−4)} \]