How do you use the Binomial Theorem to expand $(d+3)^7$ ?

Question

2685 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-07-04T19:11:29

$= d^7 + 21d^6 + 189d^5 + 945d^4 + 2835d^3 + 5103d^2 + 5103d + 2187$

$(x+y)^n = sum_(k=0)^n ((n),(k))x^(n-k)y^k$

where $((n),(k)) = (n!)/(k!(n-k)!)$ taking note that $0! = 1$

$(d+3)^7 = sum_(k=0)^7 ((7),(k))d^(7-k)3^k$

$= ((7),(0))d^(7)3^0 + ((7),(1))d^(6)3^1+((7),(2))d^(5)3^2... ((7),(7))d^(0)3^7$

$= (7!)/(0!7!)d^7 + (7!)/(1!6!)d^(6)*3 + (7!)/(2!5!)d^5*9 ... (7!)/(7!0!)*3^7$

$= d^7 + 21d^6 + 189d^5 + 945d^4 + 2835d^3 + 5103d^2 + 5103d + 2187$

Note, for brevity I have skipped out a chunk of the calculation but the method is the exact same so you should have no issue with reproducing it

How do you use the Binomial Theorem to expand (d+3)^7(d+3)^7?