factor (x-1)^3
I don't think you mean factor. I think you mean expand.
I think that this is the simple way: making it (x -1)(x-1)(x-1) and then multiply the first two parentheses to get (x2 -1x -1x +1)(x -1). You don't have to but simplify the first set of parentheses to get
(x2-2x +1)(x -1). Then multiply each term of the new first set of parentheses with each term in the second parentheses. Your result is x3 - 1x2 - 2x2 +2x +1x - 1. Then simplify, x3 -3x2 +3x - 1
There is another way which is simple but the student doesn't have to know much math at all.
Look at our answer: x3- 3x2 +3x - 1 this can be rewritten as
1x310 -3x211 +3x112 - 1x013.
Looking at just the exponents notice how the x's exponents are going from 3 to 0(left to right) and the 1's exponents are going from 0 to 3(left to right).
They each go to 3 because our original exponent was 3:(x -1)3
We use x's and 1's because they were the 2 terms in the original (x -1)3
Now look at the coefficients, they go 1 3 3 1 this is the same list of numbers in Pascal's Triangle's 4th row.
Pascal's triangle: 1
1 1
1 2 1
1 3 3 1 <------Fourth row
1 4 6 4 1
Also the terms of the answer switch from positive to negative every other. (this only happens if there is a minus in the original, if the original is a PLUS, then ALL terms of the answers are positive
We can use this technique for any type of binomial to an exponent.
Example: (x + 2)4
Coefficients: Look at the 5th row of Pascals triangle: 1 4 6 4 1
Each coefficient will be paired with an x and a 2. the exponents for x will go from 0 to 4 and the exponents for 2 will go from 4 to 0. Every term will be positive.
1x420 + 4x321 + 6x222 + 4x123 + 1x024
Now simplify your powers and coefficients: x4 + 4x32 + 6x24 + 4x8 + 16,
also simplify the multiplications: x4 + 8x3 + 24x2 + 32x + 16