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

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

Explanation:

Given expression,

`(x-5)^5`

`=(x+(-5))^5`

∵ By the binomial expansion,

`(a+b)^n=\sum_(r=0)^(n)^nC_r a^(n-r) b^(r)`

Where,

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

Thus,

`(x+(-5))^5=^5C_0 (x)^5 (-5)^(0)+^5C_1 (x)^4 (-5)^(1)+^5C_2 (x)^3 (-5)^(2)+^5C_3 (x)^2 (-5)^(3)+^5C_4 (x)^1 (-5)^(4)+^5C_0 (x)^0 (-5)^(5)`

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

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

Answer 2

I have my notes here that might help you solve the problem on your own:

We sometimes need to expand binomials as follows:

(a + b)0 = 1

(a + b)1 = a + b

(a + b)2 = a2 + 2ab + b2

(a + b)3 = a3 + 3a2b + 3ab2 + b3

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

(a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5

Clearly, doing this by direct multiplication gets quite tedious and can be

rather difficult for larger powers or more complicated expressions.