To factor the polynomial \( x^3 - 4x^2 - 4x + 16 \) by grouping, we can follow these steps:
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Group the terms: We can split the polynomial into two groups: \[ (x^3 - 4x^2) + (-4x + 16) \]
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Factor out the common factor from each group:
- In the first group \( x^3 - 4x^2 \), the common factor is \( x^2 \): \[ x^2(x - 4) \]
- In the second group \( -4x + 16 \), the common factor is \(-4\): \[ -4(x - 4) \]
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Combine the factored groups: Now we can rewrite the expression using the factored form: \[ x^2(x - 4) - 4(x - 4) \]
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Factor out the common binomial factor: Since both terms contain the factor \((x - 4)\), we can factor that out: \[ (x - 4)(x^2 - 4) \]
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Notice that \(x^2 - 4\) is a difference of squares, which can be factored further: \[ x^2 - 4 = (x - 2)(x + 2) \]
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Combine everything: Putting it all together, we have: \[ (x - 4)(x - 2)(x + 2) \]
Thus, the factored form of the polynomial \( x^3 - 4x^2 - 4x + 16 \) is: \[ \boxed{(x - 4)(x - 2)(x + 2)} \]