Answer:
C
Step-by-step explanation:
p³ - 216q³ ← is a difference of cubes and factors in general as
a³ - b³ = (a - b)(a² + ab + b²) , then
p³ - 216q³
= p³ - (6q)³
= (p - 6q)(p² + 6pq + 36q²) → C
Answer:
C. (p - 6q)(p² + 6pq + 36q²)
Step-by-step explanation:
Equation at the end of step 1
[tex](p^{3} ) - (2^{3} 3^{3} q^{3} )[/tex]
Trying to factor as a Difference of Cubes:
Factoring: p^3 - 216q^3
Theory: A difference of two perfect cubes, a^3 - b^3 can be factored into
(a-b) • (a^2 +ab +b^2)
Proof: [tex](a-b)(a^{2} +ab+b^{2} ) =[/tex]
[tex]a^{3} + a^{2} b+ab^{2} -ba^{2} -b^{2} a-b^{3} =[/tex]
[tex]a^{3} +(a^{2} b-ba^{2} )+(ab^{2} -b^{2} a)-b^{3} =[/tex]
[tex]a^{3} +0+0-b^{3} =[/tex]
[tex]a^{3} -b^{3}[/tex]
Check: 216 is the cube of 6
Check: p^3 is the cube of p^1
Check: q^3 is the cube of q^1
Factorization is:
[tex](p-6q)(p^{2} + 6pq+36q^{2} )[/tex]
Trying to factor a multi variable polynomial :
Factoring: p^2 + 6pq + 36q^2
Try to factor this multi-variable trinomial using trial and error
Factorization fails
Final Result:
(p - 6q)(p² + 6pq + 36q²)
