#### Question

Find the coefficient of x^{49} in the polynomial p(x)=(x–1)(x–2)(x–3) \dots(x-50)

Question

Find the coefficient of x^{49} in the polynomial p(x)=(x–1)(x–2)(x–3) \dots(x-50)

(x-a_1)(x-a_2)(x-a_3)

=[x^2-(a_1+a_2)x+a_1a_2)](x-a_3)

=x^3-(a_1+a_2)x^2+a_1a_2x-a_3x^2 +(a_1+a_2)a_3x-a_1a_2a_3

=x^3-(a_1+a_2+a_3)x^2+(a_1a_2+a_1a_3+a_1a_3)x-a_1a_2a_3

So the coefficient of the term x^{49} is,

-(1+2+3+\dots+50)

=\dfrac{51}{2} \times 50

=50\times 25+25

=1275

Using vieta’s formula, the second leading term has the coefficient that is equal to the opposite of sum of all roots.

Therefore,the coefficient of x^{49} is

-(1+2+\dots+50)

=-1275