Question

How do you expand `( 4- x ) ^ { 4}` using the Pascal triangle?

Answer 1

`x^4 - 16x^3 + 96x^2 - 256x + 256`

Explanation:

Let us take a look at the fourth row of Pascal's triangle:
`1, 4, 6, 4, 1`

These are our coefficients in the expansion for a binomial raised to the power of four.

Going from left to right in our row of coefficients, let us call the first number the first coefficient, the second the second coefficient, and so on...

The actual terms are calculated like this. Note the pattern of increase and decrease in the exponents.

1st term = First coefficient `* (-x)^4 * (4)^0`

2nd term = Second coefficient `* (-x)^3 * (4)^1`

3rd term = Third coefficient `* (-x)^2 * (4)^2`

4th term = Fourth coefficient `* (-x)^1 * (4)^3`

5th term = Fifth coefficient `* (-x)^0 * (4)^4`

The `-x` terms' exponents start from the degree, four, and decrease all the way to zero, while the `4` terms' exponents go the opposite way, that is the exponents start at zero and end at the degree, four.

The expansion, putting all this information together, will look like this:

`(4-x)^4 = (1*(-x)^4*4^0) + (4*(-x)^3*4^1) + (6*(-x)^2*4^2) + (4*(-x)^1*4^3) + (1*(-x)^0*4^4)`

Positive exponents yield positive results, and the zeroth power evaluates to 1,

`=x^4 - 16x^3 + 96x^2 - 256x + 256`