Question

factorise 27x^3 - y^3/27+8z^3+6xyz​

Answer 1

Step by step solution :

Step 1 :

Equation at the end of step 1 :

(y3)
(((27•(x3))-————)+23z3)+6xyz
27
Step 2 :

y3
Simplify ——
27
Equation at the end of step 2 :

y3
(((27 • (x3)) - ——) + 23z3) + 6xyz
27
Step 3 :

Equation at the end of step 3 :

y3
((33x3 - ——) + 23z3) + 6xyz
27
Step 4 :

Rewriting the whole as an Equivalent Fraction :

4.1 Subtracting a fraction from a whole

Rewrite the whole as a fraction using 27 as the denominator :

33x3 33x3 • 27
33x3 = ———— = —————————
1 27
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

4.2 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

33x3 • 27 - (y3) 729x3 - y3
———————————————— = ——————————
27 27
Equation at the end of step 4 :

(729x3 - y3)
(———————————— + 23z3) + 6xyz
27
Step 5 :

Rewriting the whole as an Equivalent Fraction :

5.1 Adding a whole to a fraction

Rewrite the whole as a fraction using 27 as the denominator :

23z3 23z3 • 27
23z3 = ———— = —————————
1 27
Trying to factor as a Difference of Cubes:

5.2 Factoring: 729x3 - y3

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 : 729 is the cube of 9

Check : x3 is the cube of x1

Check : y3 is the cube of y1

Factorization is :
(9x - y) • (81x2 + 9xy + y2)

Trying to factor a multi variable polynomial :

5.3 Factoring 81x2 + 9xy + y2

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

Factorization fails

Adding fractions that have a common denominator :

5.4 Adding up the two equivalent fractions

(9x-y) • (81x2+9xy+y2) + 23z3 • 27 729x3 - y3 + 216z3
—————————————————————————————————— = ——————————————————
27 27
Equation at the end of step 5 :

(729x3 - y3 + 216z3)
———————————————————— + 6xyz
27
Step 6 :

Rewriting the whole as an Equivalent Fraction :

6.1 Adding a whole to a fraction

Rewrite the whole as a fraction using 27 as the denominator :

6xyz 6xyz • 27
6xyz = ———— = —————————
1 27
Trying to factor a multi variable polynomial :

6.2 Factoring 729x3 - y3 + 216z3

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

Factorization fails

Adding fractions that have a common denominator :

6.3 Adding up the two equivalent fractions

(729x3-y3+216z3) + 6xyz • 27 729x3 + 162xyz - y3 + 216z3
———————————————————————————— = ———————————————————————————
27 27
Checking for a perfect cube :

6.4 729x3 + 162xyz - y3 + 216z3 is not a perfect cube

Final result :

729x3 + 162xyz - y3 + 216z3
———————————————————————————
27