Solve this: Find the coefficient of x49 in the polynomial x-C1C0x-22C2C1x-32 C3C2 - Maths - Binomial Theorem

Question

Solve this:
Find the coefficient of x49 in the polynomial
x - C 1 C 0 x - 2 2 C 2 C 1 x - 3 2 C 3 C 2 x - 50 2 C 50 C 49
  w h e r e   C r = C r 50

Neha Sethi · Accepted Answer

Dear student
We have,x-C1C0x-22C2C1x-32C3C2
x-502C50C49Let the polynomial is x50+A49 x49+A48x48+
A1x+A0=0whose roots are C1C0,22C2C1,32C3C2,
,502C50C49So, sum of roots=-Coeff of x49Coeff of x50=-A49A49=-C1C0+22C2C1+32C3C2+
+502C50C49=∑r=150-r2CrCr-1=∑r=150r2-51r=∑r=150r2-51∑r=150r=5050+12×50+16-515050+12    ∵∑n=1kn2=n(n+1)(2n+1)6 and ∑n=1kn=nn+12 =50×51×1016-51×50×512=50×5121013-51=25502-523=-22100
Regards

Solve this: Find the coefficient of x49 in the polynomial x - C 1 C 0 x - 2 2 C 2 C 1 x - 3 2 . C 3 C 2 . . . . . . . . . . . . . . . . x - 50 2 . C 50 C 49 w h e r e C r = C r 50 .

Solve this:
Find the coefficient of x⁴⁹ in the polynomial
$(x - \frac{C_{1}}{C_{0}}) (x - 2^{2} \frac{C_{2}}{C_{1}}) (x - 3^{2} . \frac{C_{3}}{C_{2}}) . . . . . . . . . . . . . . . . (x - 50^{2} . \frac{C_{50}}{C_{49}}) w h e r e C_{r} = {}^{50}C_{r} .$