Final answer:
The expression 16x² - 16xy³ + 4y⁶ is factored by first finding the greatest common factor, which is 4, and then recognizing that the resulting terms form a perfect square trinomial. The completely factored expression is 4(2x - y³)².
Step-by-step explanation:
The student has asked to factor the expression: 16x² - 16xy³ + 4y⁶. To factor this expression, we look for a greatest common factor (GCF) that each term of the polynomial shares. The GCF for this expression is 4, so we can factor out 4 from each term:
4(4x² - 4xy³ + y⁶)
Now we can factor by grouping since we have a four-term polynomial. But we need to recognize that the remaining expression after factoring out the GCF is actually a perfect square trinomial. The terms (2x)², -2(2x)(y³), and (y³)² are the squares and cross product of (2x - y³)². So, the fully factored form is:
4(2x - y³)²