If H.C.F and L.C.M of two polynomials are x - 3 and x³ - 9x² + 26x - 24 respectively. Find the second polynomial when polynomial is x² - 5x - 6?

Answer: x² - 7x + 12

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

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