Sure, let's factorize the given polynomial step-by-step.
Given polynomial:
[tex]\[ 125p^3 - 8q^3 \][/tex]
We recognize that this is a difference of cubes. The formula for the difference of cubes is:
[tex]\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \][/tex]
We need to identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that [tex]\(a^3\)[/tex] and [tex]\(b^3\)[/tex] match the given terms in the polynomial. Comparing [tex]\(125p^3\)[/tex] and [tex]\(8q^3\)[/tex] to the cube formula:
[tex]\[a = 5p\][/tex]
[tex]\[b = 2q\][/tex]
Plugging [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the factorization formula:
[tex]\[ (5p - 2q)((5p)^2 + (5p)(2q) + (2q)^2) \][/tex]
Now, let's simplify the second factor:
- [tex]\((5p)^2 = 25p^2\)[/tex]
- [tex]\((5p)(2q) = 10pq\)[/tex]
- [tex]\((2q)^2 = 4q^2\)[/tex]
Putting it all together, we get:
[tex]\[ 125p^3 - 8q^3 = (5p - 2q)(25p^2 + 10pq + 4q^2) \][/tex]
So the fully factorized form of the polynomial [tex]\( 125p^3 - 8q^3 \)[/tex] is:
[tex]\[ (5p - 2q)(25p^2 + 10pq + 4q^2) \][/tex]
Therefore, the quadratic polynomial that completes the factorization is:
[tex]\[ 25p^2 + 10pq + 4q^2 \][/tex]
