Question

factor completely:
$w^{3}x^{4}-16w^{3}$.

Explanation:

Step1: Factor out the GCF

The greatest common factor (GCF) of \(w^{3}x^{4}\) and \(16w^{3}\) is \(w^{3}\). Factor out \(w^{3}\):
\(w^{3}x^{4}-16w^{3}=w^{3}(x^{4} - 16)\)

Step2: Factor the difference of squares

Notice that \(x^{4}-16\) is a difference of squares, since \(x^{4}=(x^{2})^{2}\) and \(16 = 4^{2}\). Use the formula \(a^{2}-b^{2}=(a + b)(a - b)\) where \(a=x^{2}\) and \(b = 4\):
\(x^{4}-16=(x^{2}+4)(x^{2}-4)\)

Step3: Factor the remaining difference of squares

Now, \(x^{2}-4\) is also a difference of squares, with \(a=x\) and \(b = 2\). Apply the difference of squares formula again:
\(x^{2}-4=(x + 2)(x - 2)\)

Step4: Combine all factors

Substitute the factored forms back into the expression:
\(w^{3}x^{4}-16w^{3}=w^{3}(x^{2}+4)(x + 2)(x - 2)\)

Answer:

\(w^{3}(x^{2}+4)(x + 2)(x - 2)\)