Question

How do I factor 125x^3+8

Answer 1

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)