To find the quadratic polynomial that completes the factorization of the given cubic polynomial [tex]\( 512p^3 + 125q^3 \)[/tex], we can start from the factorization format given:
[tex]\[
512 p^3 + 125 q^3 = (8p + 5q)(\square)
\][/tex]
We aim to determine what replaces the [tex]\(\square\)[/tex] to complete the factorization of the expression. Let's denote this missing polynomial as [tex]\( A \)[/tex], so our equation becomes:
[tex]\[
512 p^3 + 125 q^3 = (8p + 5q)A
\][/tex]
Since the left-hand side of the equation is a cubic expression, [tex]\(A\)[/tex] must be a quadratic polynomial because the highest power term on the left side is [tex]\( p^3 \)[/tex].
Next, let's assume [tex]\( A \)[/tex] to be of the form:
[tex]\[
A = ap^2 + bpq + cq^2
\][/tex]
Therefore,
[tex]\[
(8p + 5q)(ap^2 + bpq + cq^2)
\][/tex]
To find [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], we will expand and equate the coefficients with the original polynomial on the left-hand side:
Expanding the right-hand side:
[tex]\[
(8p + 5q)(ap^2 + bpq + cq^2) = 8p(ap^2) + 8p(bpq) + 8p(cq^2) + 5q(ap^2) + 5q(bpq) + 5q(cq^2)
\][/tex]
This expands to:
[tex]\[
8ap^3 + 8bp^2q + 8cpq^2 + 5ap^2q + 5bpq^2 + 5cq^3
\][/tex]
Combining like terms, we get:
[tex]\[
8ap^3 + (8b+5a)p^2q + (8c+5b)pq^2 + 5cq^3
\][/tex]
Given that this must equal [tex]\( 512p^3 + 125q^3 \)[/tex], we compare coefficients of like terms. We have:
1. Coefficient of [tex]\( p^3 \)[/tex]:
[tex]\[
8a = 512 \implies a = \frac{512}{8} = 64
\][/tex]
2. Coefficient of [tex]\( q^3 \)[/tex]:
[tex]\[
5c = 125 \implies c = \frac{125}{5} = 25
\][/tex]
3. Coefficient of [tex]\( p^2q \)[/tex] and [tex]\( pq^2 \)[/tex]:
[tex]\[
8b + 5a = 0 \quad \text{and} \quad 8c + 5b = 0
\][/tex]
Let's solve for [tex]\(b\)[/tex]:
From [tex]\((8b + 5a = 0)\)[/tex]:
[tex]\[
8b + 5(64) = 0 \implies 8b + 320 = 0 \implies 8b = -320 \implies b = -40
\][/tex]
Thus, substituting [tex]\( a = 64 \)[/tex], [tex]\(b = -40\)[/tex], and [tex]\( c = 25 \)[/tex] into the assumed form of [tex]\( A \)[/tex], we get:
[tex]\[
A = 64p^2 - 40pq + 25q^2
\][/tex]
So, the quadratic polynomial that completes the factorization is:
[tex]\[
64p^2 - 40pq + 25q^2
\][/tex]
Therefore,
[tex]\[
512 p^3 + 125 q^3 = (8 p + 5 q)(64 p^2 - 40 p q + 25 q^2)
\][/tex]
Hence, the quadratic polynomial is [tex]\(\boxed{64 p^2 - 40 p q + 25 q^2}\)[/tex].
