Which of the following is the correct factorization of the polynomial below?

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)

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