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Answer :
To factor out the greatest common factor (GCF) from the polynomial [tex]\(27x^2 + 81x^8 - 18x^4 + 45\)[/tex], let's follow these steps:
1. Identify the GCF of the coefficients:
- Look at the coefficients: 27, 81, 18, and 45.
- List the factors of each:
- 27: 1, 3, 9, 27
- 81: 1, 3, 9, 27, 81
- 18: 1, 2, 3, 6, 9, 18
- 45: 1, 3, 5, 9, 15, 45
- The greatest common factor of these numbers is 9.
2. Look at the variable parts:
- The variable terms involve powers of [tex]\(x\)[/tex]: [tex]\(x^2\)[/tex], [tex]\(x^8\)[/tex], and [tex]\(x^4\)[/tex].
- Since not every term has an [tex]\(x\)[/tex], the variable [tex]\(x\)[/tex] itself cannot be factored out as part of the GCF.
3. Factor out the GCF:
- Now, factor out the GCF of 9 from each term:
[tex]\[
27x^2 + 81x^8 - 18x^4 + 45 = 9(3x^2 + 9x^8 - 2x^4 + 5)
\][/tex]
The factored form of the polynomial, with the GCF factored out, is [tex]\(9(3x^2 + 9x^8 - 2x^4 + 5)\)[/tex].
1. Identify the GCF of the coefficients:
- Look at the coefficients: 27, 81, 18, and 45.
- List the factors of each:
- 27: 1, 3, 9, 27
- 81: 1, 3, 9, 27, 81
- 18: 1, 2, 3, 6, 9, 18
- 45: 1, 3, 5, 9, 15, 45
- The greatest common factor of these numbers is 9.
2. Look at the variable parts:
- The variable terms involve powers of [tex]\(x\)[/tex]: [tex]\(x^2\)[/tex], [tex]\(x^8\)[/tex], and [tex]\(x^4\)[/tex].
- Since not every term has an [tex]\(x\)[/tex], the variable [tex]\(x\)[/tex] itself cannot be factored out as part of the GCF.
3. Factor out the GCF:
- Now, factor out the GCF of 9 from each term:
[tex]\[
27x^2 + 81x^8 - 18x^4 + 45 = 9(3x^2 + 9x^8 - 2x^4 + 5)
\][/tex]
The factored form of the polynomial, with the GCF factored out, is [tex]\(9(3x^2 + 9x^8 - 2x^4 + 5)\)[/tex].
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