Question

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

Answer 1

The correct factorization is

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

Step-by-step 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 * 7q + ({7q})^(2) )`

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