Let x R be so small that the powers of x beyond two are insignificant and negligibly small. For such x , if (1-x)^3(2+x)^6 is approximated by a+b x+c x^2 , then a+b+c=

Step-by-step solution

We have, gathered (1-x)^3=1-3 x+3 x^2 [neglecting higher term] array c (2+x)^6= ^6 C₀ 2^6 x^0+ ^6 C₁ 2^5 x^1+ ^6 C₂ 2^4 x^2 [neglecting higher term] array =64+192 x+240 x^2 gathered [neglecting higher term] =64+192 x+240 x^2 Now, aligned & aligned &(1-x)^3(2+x)^6= (1-3 x+3 x^2 ) (64+192 x+240 x^2 ) &= 64+192 x+240 x^2-192 x-576 x^2+192 x^2 & (neglecting higher term) &=-144 x^2+64 & a=64, b=0, c=-144 & a+b+c=64+0-144=-80 aligned aligned