Question

factor completely:
$v^{4}-v^{4}x^{4}$.

Explanation:

Step 1: Factor out the greatest common factor (GCF)

The GCF of \(v^{4}\) and \(v^{4}x^{4}\) is \(v^{4}\). So we factor out \(v^{4}\) from the expression:
\(v^{4}-v^{4}x^{4}=v^{4}(1 - x^{4})\)

Step 2: Factor the difference of squares

Notice that \(1 - x^{4}\) is a difference of squares, since \(1=(1)^{2}\) and \(x^{4}=(x^{2})^{2}\). The formula for factoring a difference of squares is \(a^{2}-b^{2}=(a + b)(a - b)\). Here, \(a = 1\) and \(b=x^{2}\), so:
\(1 - x^{4}=(1 + x^{2})(1 - x^{2})\)

Step 3: Factor the remaining difference of squares

Now, \(1 - x^{2}\) is also a difference of squares, with \(a = 1\) and \(b = x\). Using the difference of squares formula again:
\(1 - x^{2}=(1 + x)(1 - x)\)

Step 4: Combine all the factors

Putting it all together, we substitute back the factored forms:
\(v^{4}(1 - x^{4})=v^{4}(1 + x^{2})(1 + x)(1 - x)\)

Answer:

\(v^{4}(1 + x^{2})(1 + x)(1 - x)\)