Skip to main contentSkip to question navigationSkip to solution

IPMAT Indore 2022 (SA) PYQs

IPMAT Indore 2022

Modern Math
>
Binomial Theorem

Medium

The sum of the coefficients of all the terms in the expansion of (5x9)4(5 x-9)^{4} is __________.

Entered answer:

Correct Answer: 256
To find the sum of coefficients in any polynomial expansion, substitute x=1x = 1.
Take the original expression: (5x9)4(5x - 9)^4
Substitute x=1x = 1:
(5(1)9)4=(59)4=(4)4(5(1) - 9)^4 = (5 - 9)^4 = (-4)^4
Calculate (4)4(-4)^4:
(4)4=(4)×(4)×(4)×(4)=16×16=256(-4)^4 = (-4) \times (-4) \times (-4) \times (-4) = 16 \times 16 = 256
When we expand (5x9)4(5x - 9)^4, we get: ax4+bx3+cx2+dx+eax^4 + bx^3 + cx^2 + dx + e
To find a+b+c+d+ea + b + c + d + e, substitute x=1x = 1:
a(1)4+b(1)3+c(1)2+d(1)+e=a+b+c+d+ea(1)^4 + b(1)^3 + c(1)^2 + d(1) + e = a + b + c + d + e
This method works for any polynomial and is much faster than full expansion.
\therefore Sum of coefficients =256= 256

Keyboard Shortcuts

  • Left arrow: Previous question
  • Right arrow: Next question
  • S key: Jump to solution
  • Q key: Jump to question