Final answer:
The expression 64x⁴ + xy³ is factored by first taking out the common factor of 'x' and then recognizing the sum of cubes, resulting in x(4x + y)(16x² - 4xy + y²) as the final factored form.
Step-by-step explanation:
The student's question is asking to factor completely the algebraic expression 64x⁴ + xy³. To factor this expression, we first look for any common factors between the terms. In this case, there is just one common factor that's evident, which is 'x'. We then factor out 'x' from the expression:
x(64x³ + y³)
Next, we apply the concept of Integer Powers to see if there's a further factoring that can be done with the resulting terms. The term 64x³ can be seen as (4x)³, a cube of 4x since 4³ = 64 and (x³) is just the cube of x. Similarly, y³ is a cube in itself. We now see that we have a sum of cubes, which can be factored according to the sum of cubes formula a³ + b³ = (a + b)(a² - ab + b²). Here, 'a' is 4x and 'b' is y. Applying the formula results in:
x[(4x + y)((4x)² - (4x)(y) + y³)]
This gives us the final factored form:
x(4x + y)(16x² - 4xy + y²)
We were able to factor completely the given expression without needing to multiply both sides or using any advanced factoring techniques as mentioned in the reference provided.