Question

125x^6 +y^6 factorise using sum or difference of two cubes

Answer 1

Step-by-step explanation:

(5x2 - y2) • (25x4 + 5x2y2 + y4)

Step by step solution :

Step 1 :

Equation at the end of step 1 :

53x6 - y6

Step 2 :

Trying to factor as a Difference of Squares :

2.1 Factoring: 125x6-y6

Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)

Proof : (A+B) • (A-B) =

A2 - AB + BA - B2 =

A2 - AB + AB - B2 =

A2 - B2

Note : AB = BA is the commutative property of multiplication.

Note : - AB + AB equals zero and is therefore eliminated from the expression.

Check : 125 is not a square !!

Ruling : Binomial can not be factored as the

difference of two perfect squares

Trying to factor as a Difference of Cubes:

2.2 Factoring: 125x6-y6

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 : x6 is the cube of x2

Check : y6 is the cube of y2

Factorization is :

(5x2 - y2) • (25x4 + 5x2y2 + y4)

Trying to factor as a Difference of Squares :

2.3 Factoring: 5x2 - y2

Check : 5 is not a square !!

Ruling : Binomial can not be factored as the

difference of two perfect squares

Trying to factor a multi variable polynomial :

2.4 Factoring 25x4 + 5x2y2 + y4

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

Factorization fails

Final result :

(5x2 - y2) • (25x4 + 5x2y2 + y4)