Answer :
To find the greatest common factor (GCF) of the polynomial [tex]\(25x^{12} - 15x^9 + 10x^6\)[/tex], we need to break down the problem into manageable steps. Here’s how to do it:
1. Factor each term:
- The first term is [tex]\(25x^{12}\)[/tex].
- [tex]\(25\)[/tex] is [tex]\(5^2\)[/tex].
- So, the term is [tex]\(5^2 x^{12}\)[/tex].
- The second term is [tex]\(-15x^9\)[/tex].
- [tex]\(15\)[/tex] is [tex]\(3 \times 5\)[/tex].
- So, the term is [tex]\(-3 \times 5 \times x^9 = -5 \times 3 \times x^9 \)[/tex].
- The third term is [tex]\(10x^6\)[/tex].
- [tex]\(10\)[/tex] is [tex]\(2 \times 5\)[/tex].
- So, the term is [tex]\(2 \times 5 \times x^6 = 5 \times 2 \times x^6\)[/tex].
2. Identify the common factors:
- Look for factors that are common to each term:
- Numerical coefficient: The common factor of the numbers (25, 15, 10) is [tex]\(5\)[/tex].
- Variable [tex]\(x\)[/tex]: The lowest power of [tex]\(x\)[/tex] common to all terms is [tex]\(x^6\)[/tex].
3. Combine the identified factors:
- The greatest common factor from the coefficients is [tex]\(5\)[/tex].
- The greatest common factor from the powers of [tex]\(x\)[/tex] is [tex]\(x^6\)[/tex].
4. Conclusion:
- Therefore, the GCF of the polynomial [tex]\(25x^{12} - 15x^9 + 10x^6\)[/tex] is [tex]\(5x^6\)[/tex].
This step-by-step method leads us to the GCF of the polynomial, which can help simplify or factor the polynomial further if needed.
1. Factor each term:
- The first term is [tex]\(25x^{12}\)[/tex].
- [tex]\(25\)[/tex] is [tex]\(5^2\)[/tex].
- So, the term is [tex]\(5^2 x^{12}\)[/tex].
- The second term is [tex]\(-15x^9\)[/tex].
- [tex]\(15\)[/tex] is [tex]\(3 \times 5\)[/tex].
- So, the term is [tex]\(-3 \times 5 \times x^9 = -5 \times 3 \times x^9 \)[/tex].
- The third term is [tex]\(10x^6\)[/tex].
- [tex]\(10\)[/tex] is [tex]\(2 \times 5\)[/tex].
- So, the term is [tex]\(2 \times 5 \times x^6 = 5 \times 2 \times x^6\)[/tex].
2. Identify the common factors:
- Look for factors that are common to each term:
- Numerical coefficient: The common factor of the numbers (25, 15, 10) is [tex]\(5\)[/tex].
- Variable [tex]\(x\)[/tex]: The lowest power of [tex]\(x\)[/tex] common to all terms is [tex]\(x^6\)[/tex].
3. Combine the identified factors:
- The greatest common factor from the coefficients is [tex]\(5\)[/tex].
- The greatest common factor from the powers of [tex]\(x\)[/tex] is [tex]\(x^6\)[/tex].
4. Conclusion:
- Therefore, the GCF of the polynomial [tex]\(25x^{12} - 15x^9 + 10x^6\)[/tex] is [tex]\(5x^6\)[/tex].
This step-by-step method leads us to the GCF of the polynomial, which can help simplify or factor the polynomial further if needed.
