Question

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

Answer 1

Answer:

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

Step-by-step 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,

$^nC_r=\frac{n!}{(n-r)!r!}$

Thus, by the above formula,

$(x-5)^5=\sum_{r=0}^{5} ^5C_rx^{5-r} (-5)^r$

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

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

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

Answer 2

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