Final answer:
The polynomial 15x³ - 10x²y - 15x²y² + 10xy³ is completely factored by first factoring out the GCF, which is 5x, and then rearranging and factoring the grouped terms to obtain 5x(x + y²)(3x - 2y).
Step-by-step explanation:
To factor the polynomial 15x³ - 10x²y - 15x²y² + 10xy³ completely, we can seek out the greatest common factor (GCF) of each term and then use it to simplify the polynomial.
First, we identify the GCF of the coefficients: the numbers 15, -10, -15, and 10. The GCF here is 5.
Next, we look at the variables. The lowest power of x in the terms is x, and y does not appear in all terms, so we cannot factor it out. Thus, the GCF is 5x.
We then divide each term by this GCF:
- (15x³) / (5x) = 3x²
- (-10x²y) / (5x) = -2xy
- (-15x²y²) / (5x) = -3xy²
- (10xy³) / (5x) = 2y³
After factoring out 5x, the polynomial can be rewritten as:
5x(3x² - 2xy - 3xy² + 2y³)
Looking at the remaining expression in the parentheses, we can group the terms in pairs: (3x² - 2xy) and (-3xy² + 2y³). Within each pair, we can factor again.
- (3x² - 2xy) factored by x: x(3x - 2y)
- (-3xy² + 2y³) factored by y²: y²(-3x + 2y)
Note that the expressions (3x - 2y) and (-3x + 2y) are negatives of each other and can be written as (3x - 2y) and -(3x - 2y). Combining everything, the completely factored polynomial is:
5x(x + y²)(3x - 2y)