Questions: Factor completely: 32v^4 - 2v^4w^4.

Factor completely: 32v^4 - 2v^4w^4.
Transcript text: Factor completely: \[ 32 v^{4}-2 v^{4} w^{4} \text {. } \]
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Solution

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Solution Steps

To factor the expression completely, first identify the greatest common factor (GCF) of the terms. Then, factor out the GCF and simplify the expression further if possible.

Step 1: Identify the Greatest Common Factor (GCF)

The given expression is \(32v^4 - 2v^4w^4\). The GCF of the terms is \(2v^4\).

Step 2: Factor Out the GCF

Factor out \(2v^4\) from the expression: \[ 32v^4 - 2v^4w^4 = 2v^4(16 - w^4) \]

Step 3: Factor the Difference of Squares

The expression \(16 - w^4\) is a difference of squares: \[ 16 - w^4 = (4^2 - (w^2)^2) = (4 - w^2)(4 + w^2) \]

Step 4: Further Factor the Difference of Squares

The term \(4 - w^2\) is also a difference of squares: \[ 4 - w^2 = (2 - w)(2 + w) \]

Step 5: Combine All Factors

Combine all the factors: \[ 32v^4 - 2v^4w^4 = 2v^4(2 - w)(2 + w)(4 + w^2) \]

Final Answer

The completely factored form of the expression is: \[ \boxed{2v^4(2 - w)(2 + w)(4 + w^2)} \]

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