Answer :
To find the greatest common monomial factor of the given polynomial [tex]\(9x^4y^3 + 81x^7y^5 - 72x^2y^6\)[/tex], and to express the polynomial in its factored form, follow these steps:
1. Identify the coefficients and variables in each term:
- First term: [tex]\(9x^4y^3\)[/tex]
- Second term: [tex]\(81x^7y^5\)[/tex]
- Third term: [tex]\(-72x^2y^6\)[/tex]
2. Find the greatest common factor (GCF) for the coefficients:
- Coefficients: 9, 81, and 72
- The GCF of these coefficients is 9.
3. Find the least power of each variable present in all the terms:
- For [tex]\(x\)[/tex]:
- First term: [tex]\(x^4\)[/tex]
- Second term: [tex]\(x^7\)[/tex]
- Third term: [tex]\(x^2\)[/tex]
- The smallest power is [tex]\(x^2\)[/tex].
- For [tex]\(y\)[/tex]:
- First term: [tex]\(y^3\)[/tex]
- Second term: [tex]\(y^5\)[/tex]
- Third term: [tex]\(y^6\)[/tex]
- The smallest power is [tex]\(y^3\)[/tex].
4. Combine the GCF of the coefficients with the lowest powers of the variables:
The greatest common monomial factor is [tex]\(9x^2y^3\)[/tex].
5. Factor the original polynomial using the greatest common monomial factor:
- Divide each term of the polynomial by [tex]\(9x^2y^3\)[/tex]:
- [tex]\(9x^4y^3 \div 9x^2y^3 = x^2\)[/tex]
- [tex]\(81x^7y^5 \div 9x^2y^3 = 9x^5y^2\)[/tex]
- [tex]\(-72x^2y^6 \div 9x^2y^3 = -8y^3\)[/tex]
- Write the factored form:
[tex]\(9x^2y^3(x^2 + 9x^5y^2 - 8y^3)\)[/tex]
Therefore, the greatest common monomial factor is [tex]\(9x^2y^3\)[/tex], and the factored form of the polynomial is [tex]\(9x^2y^3(x^2 + 9x^5y^2 - 8y^3)\)[/tex].
1. Identify the coefficients and variables in each term:
- First term: [tex]\(9x^4y^3\)[/tex]
- Second term: [tex]\(81x^7y^5\)[/tex]
- Third term: [tex]\(-72x^2y^6\)[/tex]
2. Find the greatest common factor (GCF) for the coefficients:
- Coefficients: 9, 81, and 72
- The GCF of these coefficients is 9.
3. Find the least power of each variable present in all the terms:
- For [tex]\(x\)[/tex]:
- First term: [tex]\(x^4\)[/tex]
- Second term: [tex]\(x^7\)[/tex]
- Third term: [tex]\(x^2\)[/tex]
- The smallest power is [tex]\(x^2\)[/tex].
- For [tex]\(y\)[/tex]:
- First term: [tex]\(y^3\)[/tex]
- Second term: [tex]\(y^5\)[/tex]
- Third term: [tex]\(y^6\)[/tex]
- The smallest power is [tex]\(y^3\)[/tex].
4. Combine the GCF of the coefficients with the lowest powers of the variables:
The greatest common monomial factor is [tex]\(9x^2y^3\)[/tex].
5. Factor the original polynomial using the greatest common monomial factor:
- Divide each term of the polynomial by [tex]\(9x^2y^3\)[/tex]:
- [tex]\(9x^4y^3 \div 9x^2y^3 = x^2\)[/tex]
- [tex]\(81x^7y^5 \div 9x^2y^3 = 9x^5y^2\)[/tex]
- [tex]\(-72x^2y^6 \div 9x^2y^3 = -8y^3\)[/tex]
- Write the factored form:
[tex]\(9x^2y^3(x^2 + 9x^5y^2 - 8y^3)\)[/tex]
Therefore, the greatest common monomial factor is [tex]\(9x^2y^3\)[/tex], and the factored form of the polynomial is [tex]\(9x^2y^3(x^2 + 9x^5y^2 - 8y^3)\)[/tex].
