Question

How to factor 125x^3 - 64

Answer 1

Explanation:

Factor 125
`x^(3)`-64

STEP 1: Equation at the end of step 1

5³ₓ³ - 64

STEP 2:

Trying to factor as a Difference of Cubes:

2.1 Factoring: 125
`x^(3)`-64

Theory : A difference of two perfect cubes
`a^3 - b^3` ,can be factored into

(a-b) • (a² +ab +b²)

Proof : (a-b)•(a²+ab+b²) =

`a^3+a^2b+ab^2-ba^2-b^2a-b^3 =`

`a^3+(a^2b-ba^2)+(ab^2-b^2a)-b^3 =`

`a^3+0+0-b^3 =`

`a^3-b^3`

Check : 125 is the cube of 5

Check : 64 is the cube of 4

Check : x³ is the cube of x¹

Factorization is :

(5
`x` - 4) • (25
`x^(2)` + 20
`x` + 16)

Trying to factor by splitting the middle term

2.2 Factoring 25
`x^(2)` + 20
`x` + 16

The first term is, 25
`x^(2)` its coefficient is 25 .

The middle term is, +20
`x` its coefficient is 20 .

The last term, "the constant", is +16

Step-1 :

Multiply the coefficient of the first term by the constant 25 • 16 = 400

Step-2 :

Find two factors of 400 whose sum equals the coefficient of the middle term, which is 20 .

-400 + -1 = -401

-200 + -2 = -202

-100 + -4 = -104

-80 + -5 = -85

-50 + -8 = -58

-40 + -10 = -50

For tidiness, printing of 24 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 - 4) • (25x^2 + 20x + 16)`

Answer 2

Factor 125x^3−64

125x^3−64 = (5x−4)(25x^2+20x+16)

Answer: (5x−4)(25x^2+20x+16)