Answer :
To factor out the greatest common factor (GCF) of the polynomial [tex]\(12x^5y^3 - 20x^4y^5 + 4x^2y^6\)[/tex], we follow these steps:
1. Identify the GCF of the coefficients:
- The coefficients are 12, -20, and 4.
- The GCF of these numbers is 4.
2. Identify the GCF of the variable [tex]\(x\)[/tex]:
- For the [tex]\(x\)[/tex] terms, we have [tex]\(x^5\)[/tex], [tex]\(x^4\)[/tex], and [tex]\(x^2\)[/tex].
- The smallest exponent for [tex]\(x\)[/tex] is 2.
- So, the GCF for [tex]\(x\)[/tex] terms is [tex]\(x^2\)[/tex].
3. Identify the GCF of the variable [tex]\(y\)[/tex]:
- For the [tex]\(y\)[/tex] terms, we have [tex]\(y^3\)[/tex], [tex]\(y^5\)[/tex], and [tex]\(y^6\)[/tex].
- The smallest exponent for [tex]\(y\)[/tex] is 3.
- So, the GCF for [tex]\(y\)[/tex] terms is [tex]\(y^3\)[/tex].
4. Combine the GCF parts:
- The overall GCF of the polynomial is [tex]\(4x^2y^3\)[/tex].
5. Factor out the GCF from each term of the polynomial:
- Start with the original polynomial: [tex]\(12x^5y^3 - 20x^4y^5 + 4x^2y^6\)[/tex].
- Factor [tex]\(4x^2y^3\)[/tex] out of each term:
- [tex]\(12x^5y^3\)[/tex] becomes [tex]\(4x^2y^3 \cdot 3x^3\)[/tex].
- [tex]\(-20x^4y^5\)[/tex] becomes [tex]\(4x^2y^3 \cdot (-5x^2y^2)\)[/tex].
- [tex]\(4x^2y^6\)[/tex] becomes [tex]\(4x^2y^3 \cdot y^3\)[/tex].
Therefore, the factored form of the polynomial is:
[tex]\[ 4x^2y^3(3x^3 - 5x^2y^2 + y^3) \][/tex]
1. Identify the GCF of the coefficients:
- The coefficients are 12, -20, and 4.
- The GCF of these numbers is 4.
2. Identify the GCF of the variable [tex]\(x\)[/tex]:
- For the [tex]\(x\)[/tex] terms, we have [tex]\(x^5\)[/tex], [tex]\(x^4\)[/tex], and [tex]\(x^2\)[/tex].
- The smallest exponent for [tex]\(x\)[/tex] is 2.
- So, the GCF for [tex]\(x\)[/tex] terms is [tex]\(x^2\)[/tex].
3. Identify the GCF of the variable [tex]\(y\)[/tex]:
- For the [tex]\(y\)[/tex] terms, we have [tex]\(y^3\)[/tex], [tex]\(y^5\)[/tex], and [tex]\(y^6\)[/tex].
- The smallest exponent for [tex]\(y\)[/tex] is 3.
- So, the GCF for [tex]\(y\)[/tex] terms is [tex]\(y^3\)[/tex].
4. Combine the GCF parts:
- The overall GCF of the polynomial is [tex]\(4x^2y^3\)[/tex].
5. Factor out the GCF from each term of the polynomial:
- Start with the original polynomial: [tex]\(12x^5y^3 - 20x^4y^5 + 4x^2y^6\)[/tex].
- Factor [tex]\(4x^2y^3\)[/tex] out of each term:
- [tex]\(12x^5y^3\)[/tex] becomes [tex]\(4x^2y^3 \cdot 3x^3\)[/tex].
- [tex]\(-20x^4y^5\)[/tex] becomes [tex]\(4x^2y^3 \cdot (-5x^2y^2)\)[/tex].
- [tex]\(4x^2y^6\)[/tex] becomes [tex]\(4x^2y^3 \cdot y^3\)[/tex].
Therefore, the factored form of the polynomial is:
[tex]\[ 4x^2y^3(3x^3 - 5x^2y^2 + y^3) \][/tex]
