Let's work through these factoring problems step-by-step.
### Problem 1:
Factor the polynomial [tex]\(5 b^2 d + 10 c^2 d^4\)[/tex] completely.
1. Identify any common factors:
- Both terms in the polynomial share a common factor of [tex]\(5d\)[/tex].
2. Factor out the greatest common factor (GCF):
- The GCF is [tex]\(5d\)[/tex].
3. Rewrite the polynomial:
[tex]\[5 b^2 d + 10 c^2 d^4 = 5d(b^2 + 2c^2 d^3)\][/tex]
So, the polynomial [tex]\(5 b^2 d + 10 c^2 d^4\)[/tex] factors completely to:
[tex]\[
5d(b^2 + 2c^2 d^3)
\][/tex]
### Problem 2:
Factor the polynomial [tex]\(2 a^3 x^4 z^5 - 6 a^3 x^3 z^6 + 3 a^6 x^2 z^6\)[/tex] completely.
1. Identify any common factors:
- All the terms share common factors of [tex]\(a^3\)[/tex], [tex]\(x^2\)[/tex], and [tex]\(z^5\)[/tex].
2. Factor out the greatest common factor (GCF):
- The GCF is [tex]\(a^3 x^2 z^5\)[/tex].
3. Rewrite the polynomial:
[tex]\[
2 a^3 x^4 z^5 - 6 a^3 x^3 z^6 + 3 a^6 x^2 z^6 = a^3 x^2 z^5 (2 x^2 - 6 x z + 3 a^3 z)
\][/tex]
So the polynomial [tex]\(2 a^3 x^4 z^5 - 6 a^3 x^3 z^6 + 3 a^6 x^2 z^6\)[/tex] factors completely to:
[tex]\[
a^3 x^2 z^5 (2 x^2 - 6 x z + 3 a^3 z)
\][/tex]
### Final Answer:
The factored forms of the given polynomials are:
1. [tex]\[
5 b^2 d + 10 c^2 d^4 = 5d (b^2 + 2c^2 d^3)
\][/tex]
2. [tex]\[
2 a^3 x^4 z^5 - 6 a^3 x^3 z^6 + 3 a^6 x^2 z^6 = a^3 x^2 z^5 (2 x^2 - 6 x z + 3 a^3 z)
\][/tex]
