Question

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

Answer 1

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

Explanation:

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

Pascal's triangle is

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`