Question

Which of the following is the correct factorization of the polynomial below?
p3 - 343q3

Answer 1

ANSWER

The correct factorization is

$p^{3} - 343q^{3} = (p - 7q)( {p}^{2} + 7pq + {49q}^{2} )$

EXPLANATION

The given expression is

${p}^{3} - 343 {q}^{3}$

We can write this as difference of cubes.

${p}^{3} - {7}^{3} {q}^{3}$

${p}^{3} - ( {7} {q})^{3}$

We can now factor using the difference of cube identity:

${x}^{3} - {y}^{3} = (x - y)( {x}^{2} + xy + {y}^{2} )$

We substitute x=p and y=7p to get:

${p}^{3} - ( {7} {q})^{3} = (p - 7q)( {p}^{2} + p \times 7q + ({7q})^{2} )$

${p}^{3} - ( {7} {q})^{3} = (p - 7q)( {p}^{2} + 7pq + {49q}^{2} )$