Explanation
Given:
HCF = x - 3
LCM = x³ - 9x² + 26x - 24
First polynomial = x² - 5x + 6 (corrected to x² - 5x - 6 doesn't seem right with given HCF, assuming it's x² - 5x + 6 for calculation purposes)
Product of two polynomials = Product of HCF and LCM
Let's denote the second polynomial as P(x).
(x² - 5x + 6) × P(x) = (x - 3) × (x³ - 9x² + 26x - 24)
Factoring LCM:
x³ - 9x² + 26x - 24 = (x - 3)(x² - 6x + 8)
Now, factoring the first polynomial:
x² - 5x + 6 = (x - 3)(x - 2)
(x - 3)(x - 2) × P(x) = (x - 3) × (x - 3)(x² - 6x + 8)
P(x) = (x - 3)(x² - 6x + 8) / (x - 2)
= (x - 3)(x - 2)(x - 4) / (x - 2)
= (x - 3)(x - 4)
= x² - 7x + 12