An expression was factorized as (x - 1)(x - 3)(x - 5) ….. (x - 99). What is the coefficient of x⁴⁹ in the expression ?

Given,

Polynomial = (x - 1)(x - 3)(x - 5) …….. (x - 99)

Here roots are 1, 3, 5, ….., 99. Let number of roots be n.

The above sequence is in an A.P. with first term (a) = 1, common difference (d) = 2 and last term (a_n) = 99

By formula,

⇒ a_n = a + (n - 1)d

⇒ 99 = 1 + 2(n - 1)

⇒ 99 - 1 = 2(n - 1)

⇒ 98 = 2(n - 1)

⇒ n - 1 = $\dfrac{98}{2}$

⇒ n - 1 = 49

⇒ n = 50.

Sum of roots = $\dfrac{n}{2}(a + l) = \dfrac{50}{2}(1 + 99) = \dfrac{50}{2} \times 100$ = 2500.

For a polynomial of the form,

(a - a₁)(x - a₂)……….

The coefficient of x^{n - 1} is the negative of the sum of all roots.

Thus, the coefficient of x⁴⁹ = -2500.

Hence, coefficient of x⁴⁹ = -2500.