Question

Expand the following using either the Binomial Theorem or Pascal’s Triangle. You must show your work for credit.

(x - 5)^5

Answer 1

the binomial coefficients are 1,5,10,10,5,1 (5/0, 5/1, 4/2, 3/3, 2/4, 1/5)

x^5-25x^4+250x^3-1250x^2+3125x-3125

Answer 2

`x^5-25x^4+250x^3-1250x^2+3125 x-3125`

Explanation:

Here the given expression is,

`(x-5)^5`

By the binomial theorem,

`(p+q)^n=\sum_(r=0)^(n) ^nC_rp^(n-r) q^r`

Where,

`^nC_r=(n!)/((n-r)!r!)`

Thus, by the above formula,

`(x-5)^5=\sum_(r=0)^(5) ^5C_rx^(5-r) (-5)^r`

`=^5C_0x^(5-0) (-5)^0 + ^5C_1x^(5-1) (-5)^1 + ^5C_2x^(5-2) (-5)^2 + ^5C_3x^(5-3) (-5)^3 + ^5C_4x^(5-4) (-5)^4 + ^5C_5x^(5-5) (-5)^5`

`=(1)(x^5)(1)+(5)(x^4)(-5)+(10)(x^3)(25)+(10)(x^2)(-125)+(5)(x^1)(625)+(1)(1)(-3125)`

`=x^5-25x^4+250x^3-1250x^2+3125 x-3125`