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Answer :
To factor the greatest common factor (GCF) out of the polynomial [tex]\(28x^{10} + 20x^9 + 4x^8\)[/tex], follow these steps:
1. Identify the coefficients: The coefficients of the polynomial are 28, 20, and 4.
2. Find the GCF of the coefficients:
- The numbers are 28, 20, and 4.
- Determine the GCF of these numbers. By considering their prime factors:
- 28 can be factored into [tex]\(2 \times 2 \times 7\)[/tex].
- 20 can be factored into [tex]\(2 \times 2 \times 5\)[/tex].
- 4 can be factored into [tex]\(2 \times 2\)[/tex].
- The number 2 appears as a factor in all three numbers, and the highest power common to all is [tex]\(2^2 = 4\)[/tex].
3. Factor out the GCF:
- Divide each term of the polynomial by the GCF (4):
- [tex]\(28x^{10} \div 4 = 7x^{10}\)[/tex]
- [tex]\(20x^9 \div 4 = 5x^9\)[/tex]
- [tex]\(4x^8 \div 4 = 1x^8\)[/tex]
- This gives you the polynomial: [tex]\(7x^{10} + 5x^9 + x^8\)[/tex].
4. Write the expression factored:
- You can now express the original polynomial as a product of the GCF and the simplified polynomial:
[tex]\[
28x^{10} + 20x^9 + 4x^8 = 4(7x^{10} + 5x^9 + x^8)
\][/tex]
Thus, the greatest common factor, 4, has been successfully factored out of the polynomial.
1. Identify the coefficients: The coefficients of the polynomial are 28, 20, and 4.
2. Find the GCF of the coefficients:
- The numbers are 28, 20, and 4.
- Determine the GCF of these numbers. By considering their prime factors:
- 28 can be factored into [tex]\(2 \times 2 \times 7\)[/tex].
- 20 can be factored into [tex]\(2 \times 2 \times 5\)[/tex].
- 4 can be factored into [tex]\(2 \times 2\)[/tex].
- The number 2 appears as a factor in all three numbers, and the highest power common to all is [tex]\(2^2 = 4\)[/tex].
3. Factor out the GCF:
- Divide each term of the polynomial by the GCF (4):
- [tex]\(28x^{10} \div 4 = 7x^{10}\)[/tex]
- [tex]\(20x^9 \div 4 = 5x^9\)[/tex]
- [tex]\(4x^8 \div 4 = 1x^8\)[/tex]
- This gives you the polynomial: [tex]\(7x^{10} + 5x^9 + x^8\)[/tex].
4. Write the expression factored:
- You can now express the original polynomial as a product of the GCF and the simplified polynomial:
[tex]\[
28x^{10} + 20x^9 + 4x^8 = 4(7x^{10} + 5x^9 + x^8)
\][/tex]
Thus, the greatest common factor, 4, has been successfully factored out of the polynomial.
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