Final answer:
To find the LCM of the polynomials x³-9x, x²-2x-15, and x²-5x, we factor each polynomial and then find the product of the highest powers of common factors. The LCM is x(x-5)(x^2-9) or x^4-14x^2+45x.
Step-by-step explanation:
The student is asking to find the least common multiple (LCM) of three polynomials: x³-9x, x²-2x-15, and x²-5x. Finding the LCM of polynomials involves factoring each polynomial completely and then taking the highest powers of common factors. Upon factoring we get:
- x³ - 9x = x(x+3)(x-3)
- x² - 2x - 15 = (x-5)(x+3)
- x² - 5x = x(x-5)
The LCM is then the product of the highest power of each factor present in any of the polynomials. Therefore, the LCM is x(x-5)(x+3)(x-3) which simplifies to x(x-5)(x^2-9) or x^4-14x^2+45x.