Absolutely, I can help you with that. Let's factor the given polynomial [tex]\( 15 a^5 x^4 y^6 + 3 a^4 x^3 y^7 - 9 a^2 x^6 y \)[/tex].
### Step-by-Step Solution
1. Identify the Greatest Common Factor (GCF):
We start by determining the GCF of the terms in the polynomial.
- Coeficients: The coefficients are [tex]\(15\)[/tex], [tex]\(3\)[/tex], and [tex]\(-9\)[/tex]. The GCF of these coefficients is [tex]\(3\)[/tex].
- a terms: The degrees of [tex]\(a\)[/tex] in each term are [tex]\(5\)[/tex], [tex]\(4\)[/tex], and [tex]\(2\)[/tex] respectively. The GCF of [tex]\(a^5\)[/tex], [tex]\(a^4\)[/tex], and [tex]\(a^2\)[/tex] is [tex]\(a^2\)[/tex] since [tex]\(a^2\)[/tex] is the highest power of [tex]\(a\)[/tex] that divides all three.
- x terms: The degrees of [tex]\(x\)[/tex] in each term are [tex]\(4\)[/tex], [tex]\(3\)[/tex], and [tex]\(6\)[/tex] respectively. The GCF of [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^6\)[/tex] is [tex]\(x^3\)[/tex] since [tex]\(x^3\)[/tex] is the highest power of [tex]\(x\)[/tex] that divides all three.
- y terms: The degrees of [tex]\(y\)[/tex] in each term are [tex]\(6\)[/tex], [tex]\(7\)[/tex], and [tex]\(1\)[/tex] respectively. The GCF of [tex]\(y^6\)[/tex], [tex]\(y^7\)[/tex], and [tex]\(y\)[/tex] is [tex]\(y\)[/tex] since [tex]\(y\)[/tex] is the highest power of [tex]\(y\)[/tex] that divides all three.
Hence, the overall GCF of the polynomial is [tex]\( 3a^2 x^3 y \)[/tex].
2. Factor out the GCF:
We now factor out [tex]\( 3a^2 x^3 y \)[/tex] from each term in the polynomial:
[tex]\[
15 a^5 x^4 y^6 + 3 a^4 x^3 y^7 - 9 a^2 x^6 y = 3a^2 x^3 y \left(\frac{15 a^5 x^4 y^6}{3a^2 x^3 y} + \frac{3 a^4 x^3 y^7}{3a^2 x^3 y} - \frac{9 a^2 x^6 y}{3a^2 x^3 y}\right)
\][/tex]
Simplify each fraction inside the parentheses:
- The first term:
[tex]\[
\frac{15 a^5 x^4 y^6}{3a^2 x^3 y} = 5 a^3 x y^5
\][/tex]
- The second term:
[tex]\[
\frac{3 a^4 x^3 y^7}{3a^2 x^3 y} = a^2 y^6
\][/tex]
- The third term:
[tex]\[
\frac{9 a^2 x^6 y}{3a^2 x^3 y} = 3 x^3
\][/tex]
Putting these simplified terms together, we get:
[tex]\[
3a^2 x^3 y \left( 5 a^3 x y^5 + a^2 y^6 - 3 x^3 \right)
\][/tex]
Thus, the factored form of the polynomial [tex]\( 15 a^5 x^4 y^6 + 3 a^4 x^3 y^7 - 9 a^2 x^6 y \)[/tex] is:
[tex]\[
3a^2 x^3 y \left( 5 a^3 x y^5 + a^2 y^6 - 3 x^3 \right)
\][/tex]
