How do you factor 125-x^3?

Question

asked Sep 25, 2023 160k views

2 Answers

JamesJJ · Answer 1 · 2023-09-29T11:15:55+0000

Explanation:

We have

$125 - x {}^(3)$

First, 125 is a perfect cube because

5 * 5 * 5 = 125

and

x^3 is a perfect cube because

$x * x * x = x {}^(3)$

so we can use the difference of cubes identity

$( {x}^(3) - {y}^(3) ) = (x - y)( {x}^(2) + xy + {y}^(2) )$

Let say we have two perfect cubes:

64 because 8×8×8=64

and 27 because 3×3×3=27 and let subtract

64 - 27

we know that

64 - 27 = 37

but using the difference of cubes identity we should get the same thing.

Remeber cube root of 64 is 4 and cube root of 27 is 3 so we have

$(4 - 3)( {4}^(2) + 4(3) + 3 {}^(2) )$

1(16 + 12 + 9) = 1(37) = 37

So the difference of cubes works for real numbers. This is a good way to help remeber the identity using real numbers.

Back on to the topic,

we know that 5 is cube root of 125 and x is the cube root of x^3 so we have

$(5 - x)( {5}^(2) + 5x + {x}^(2) ) =$

$(5 - x)(25 + 5x + {x}^(2) )$