How do you use pascals triangle to expand #(x+2)^5 #?

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Answer 1 · 2015-09-27T02:12:54

#color(red)((x+2)^5 = x^5+10x^4+40x^3+80x^2+80x+32)#

Write out the sixth row of Pascal's triangle and make the appropriate substitutions.

Pascal's triangle is

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The numbers in the sixth row are 1, 5, 10, 10, 5, 1.

They are the coefficients of the terms in a fifth order polynomial:

#(x+y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5#

But your polynomial is #(x+2)^5#.

Let #y=2#.

Then your polynomial becomes

#(x+2)^5 =(x+y)^5#.

#(x+y)^5= x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5#

Now re-insert the values of #y#.

#(x+2)^5 = x^5 + 5x^4(2) + 10x^3(2)^2 + 10x^2(2)^3 + 5x(2)^4 + 2^5#

#(x+2)^5 = x^5+5x^4×2+10x^3×4+10x^2×8+5x×16+32#

#(x+2)^5 = x^5+10x^4+40x^3+80x^2+80x+32#