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

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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#