Question

How do you factor 125-x^3?

Answer 1

Step-by-step explanation:

We have

$125 - x {}^{3}$

First, 125 is a perfect cube because

$5 \times 5 \times 5 = 125$

and

x^3 is a perfect cube because

$x \times x \times x = x {}^{3}$

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

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

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

Answer 2

First Image is the answer the second is the steps

Best of Luck