how do i factor this in a simple way factor x13SolutionI thi

how do i factor this in a simple way?
factor (x-1)^3

Solution

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