How do I factor 125x^3+8

Step 1:

53x3 - 8

Step 2:

Trying to factor as a Difference of Cubes:

2.1 Factoring: 125x3-8

Theory : A difference of two perfect cubes, a3 - b3 can be factored into

(a-b) • (a2 + ab + b2)

Proof : (a-b) • (a2+ab+b2) =

a3+a2b+ab2-ba2-b2a-b3 =

a3 + (a2b-ba2) + (ab2-b2a) - b3 =

a3+0+0+b3 =

a3+b3

Check : 125 is the cube of 5

Check : 8 is the cube of 2

Check : x3 is the cube of x1

Factorization is:

(5x - 2) • (25x2 + 10x + 4)

Trying to factor by splitting the middle term

2.2 Factoring 25x2 + 10x + 4

The first term is, 25x2 its coefficient is 25.

The middle term is, + 10x its coefficient is 10.

The last term, "the constant", is + 4

Step-1 : Multiply the coefficient of the first term by the constant 25 • 4 = 100

Step-2 : Find two factors of 100 whose sum equals the coefficient of the middle term, which is 10.

-100 + - 1 = - 101

-50 + - 2 = - 52

-25 + - 4 = - 29

-20 + - 5 = - 25

-10 + - 10 = - 20

-5 + - 20 = - 25

For tidiness, printing of 12 lines which failed to find two such factors, was suppressed

Observation : No two such factors can be found!

Conclusion : Trinomial can not be factored

Final result:

(5x - 2) • (25x2 + 10x + 4)