Question

factories =54x3 y + 250y4​

Answer 1

Explanation:

(1): "y4" was replaced by "y^4". 1 more similar replacement(s).

STEP

1

:

Equation at the end of step 1

((54 • (x3)) • y) - (2•53y4)

STEP

2

:

Equation at the end of step

2

:

((2•33x3) • y) - (2•53y4)

STEP

3

:

STEP

4

:

Pulling out like terms

4.1 Pull out like factors :

54x3y - 250y4 = 2y • (27x3 - 125y3)

Trying to factor as a Difference of Cubes:

4.2 Factoring: 27x3 - 125y3

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 : 27 is the cube of 3

Check : 125 is the cube of 5

Check : x3 is the cube of x1

Check : y3 is the cube of y1

Factorization is :

(3x - 5y) • (9x2 + 15xy + 25y2)

Trying to factor a multi variable polynomial :

4.3 Factoring 9x2 + 15xy + 25y2

Try to factor this multi-variable trinomial using trial and error

Factorization fails

Final result :

2y • (3x - 5y) • (9x2 + 15xy + 25y2)