factor (x-1)^3

## how do i factor this in a simple way?

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# 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 (x

^{2}-1x -1x +1)(x -1). You don't have to but simplify the first set of parentheses to get(x

^{2}-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 x^{3}- 1x^{2}- 2x^{2}+2x +1x - 1. Then simplify, x^{3}-3x^{2}+3x - 1There is another way which is simple but the student doesn't have to know much math at all.

Look at our answer: x

^{3}- 3x^{2}+3x - 1 this can be rewritten as**x***1*^{3}**-3x***1*^{0}^{2}**+3x***1*^{1}*-*^{1}**1**^{2}**1***1**x*^{0}**.**^{3}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.

1x

^{4}2^{0}+ 4x^{3}2^{1}+ 6x^{2}2^{2}+ 4x^{1}2^{3}+ 1x^{0}2^{4}Now simplify your powers and coefficients: x

^{4}+ 4x^{3}2 + 6x^{2}4 + 4x8 + 16,also simplify the multiplications: x

^{4}+ 8x^{3}+ 24x^{2}+ 32x + 16