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

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

Solution

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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