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

well, using the binomial theorem, is far simpler, since you get the coefficient as you go and only the ones you need.


`\bf (x-5)^5\implies \begin{array}{llll} term&coefficient&value\\ -----&-----&-----\\ 1&+1&(x)^5(-5)^0\\ 2&+5&(x)^4(-5)^1\\ 3&+10&(x)^3(-5)^2\\ 4&+10&(x)^2(-5)^3\\ 5&+5&(x)^1(-5)^4\\ 6&+1&(x)^0(-5)^5 \end{array}`

notice, with the binomial theorem, you start the first element with the highest exponent, in this case 5, and the second element with 0, and then you gradually decrease it by 1 to the first one and increase it by 1 to the second one.

That part is rather simple, now, to get the coefficient, the 1st coefficient is 1 for one, to get the next one is "the product of the current coefficient and the exponent of the first element, divided by the exponent of the second element on the next term"...which is a mouthful...but anyway ... for example

how did we get 10 for the 3rd term? 5 * 4 / 2

how did we get 10 again for the 4th term though? 10 * 3 / 3

and the 5th coefficient? 10 * 2 / 4

and so on.

so, just combine those, and that's the expansion for that binomial.

Answer 2

The expansion is
`(x-5)^5= x^5-25x^4+250x^3-1250x^2+3125 x-3125`

Explanation:

Given : Expression
`(x-5)^5`

To find : Expand the following using either the Binomial Theorem or Pascal’s Triangle?

Solution :

Applying binomial theorem,

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

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

Now, expanding
`(x-5)^5` by binomial formula,

We have,

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

Open the summation,

`\Rightarrow ^5C_0 x^(5-0) (-5)^0 + ^5C_1 x^(5-1) (-5)^1 + ^5C_2 x^(5-2) (-5)^2 + ^5C_3 x^(5-3) (-5)^3 \\+ ^5C_4x^(5-4) (-5)^4 + ^5C_5x^(5-5) (-5)^5`

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

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

Therefore, The expansion is
`(x-5)^5= x^5-25x^4+250x^3-1250x^2+3125 x-3125`