How do I use Pascal's triangle to expand the binomial #(d-5)^6#?

Question

6334 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-07-09T22:45:24

You write out the seventh row of Pascal's triangle and make the appropriate substitutions.

Pascal's triangle is

The numbers in the seventh row are 1, 6, 15, 20, 15, 6,1.

They are the coefficients of the terms in a sixth order polynomial.

#(x+y)^6 = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6#

But our polynomial is #(d-5)^6#.

Substitute #x = d# and #y = -5#.

#(d-5)^6 = d^6 + 6d^5(-5) + 15d^4(-5)^2 + 20d^3(-5)^3 + 15d^2(-5)^4 + 6d(-5)^5 + (-5)^6#

#(d-5)^6 = d^6 -30d^5 + 15× 25d^4 – 20×125d^3 + 15× 625d^2 – 6×3125d^5 + 15625#

#(d-5)^6 = d^6 -30d^5 + 375d^4 – 2500d^3 + 9375d^2 – 18750d^5 + 15625#