Use the Binomial Theorem to expand the Binomial (d-4b)^3

Question

Can you please show work? I am very lost

Answer 1

Pascal's Triangle for (a+b)ⁿ

 

n      coefficients

---------------------

0               1

1            1   1

2        1    2    1

3     1   3    3    1

 

So for (a+b)³

 

= 1(a³b⁰) + 3(a²b¹) + 3(a¹b²) + 1(a⁰b³)

 

= a³ + 3a²b + 3ab² + b³

 

The triangle gives you the numbers in front of

the variables, aka, the coefficients. The variables go

in with aⁿb⁰ + a^n-1b¹ + ... + a⁰bⁿ

 

In other words the exponents start at n for a

and decrease to zero while the exponents for b

start at zero and increase to n.

 

NOTE: a⁰ = 1

 

We have (d-4b)³

So plug into a³ + 3a²b + 3ab² + b³

the fact that a=d, b = (-4b)

 

(d-4b)³ 

= d³ +3d²(-4b) + 3d(-4b)² + (-4b)³

= d³ -12d²b + 3d(16b²) - 64b³

 

= d³ - 12d²b + 48db² - 64b³

 

Questions?  Take your time to look this over carefully.