Ask a question
0 0

how do i factor this in a simple way?

factor (x-1)^3
Tutors, please sign in to answer this question.

1 Answer

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