Answer :
To factor the expression [tex]\(-45x^8 - 27x^5 - 18x^2\)[/tex] using the greatest common factor (GCF), let's follow these steps:
1. Identify the GCF of the numerical coefficients:
- The coefficients are [tex]\(-45\)[/tex], [tex]\(-27\)[/tex], and [tex]\(-18\)[/tex].
- The GCF of these numbers is [tex]\(-9\)[/tex]. This is the largest negative number that divides all three without leaving a remainder.
2. Identify the GCF of the variable parts:
- For the variables, we have [tex]\(x^8\)[/tex], [tex]\(x^5\)[/tex], and [tex]\(x^2\)[/tex].
- The GCF of these exponents is [tex]\(x^2\)[/tex], as it is the lowest power of [tex]\(x\)[/tex] present in all terms.
3. Factor out the GCF:
- Now that we've determined the GCF to be [tex]\(-9x^2\)[/tex], we factor it out from each term in the expression:
[tex]\[
-45x^8 - 27x^5 - 18x^2 = -9x^2(5x^6 + 3x^3 + 2)
\][/tex]
4. Verify by distributing:
- To ensure the factoring is correct, distribute [tex]\(-9x^2\)[/tex]:
- Calculate [tex]\(-9x^2 \times 5x^6 = -45x^8\)[/tex]
- Calculate [tex]\(-9x^2 \times 3x^3 = -27x^5\)[/tex]
- Calculate [tex]\(-9x^2 \times 2 = -18x^2\)[/tex]
- These results match the original expression, confirming the factoring is accurate.
Thus, the expression [tex]\(-45x^8 - 27x^5 - 18x^2\)[/tex] factors to [tex]\(-9x^2(5x^6 + 3x^3 + 2)\)[/tex].
The correct answer is A: [tex]\(-9x^2(5x^6 + 3x^3 + 2)\)[/tex].
1. Identify the GCF of the numerical coefficients:
- The coefficients are [tex]\(-45\)[/tex], [tex]\(-27\)[/tex], and [tex]\(-18\)[/tex].
- The GCF of these numbers is [tex]\(-9\)[/tex]. This is the largest negative number that divides all three without leaving a remainder.
2. Identify the GCF of the variable parts:
- For the variables, we have [tex]\(x^8\)[/tex], [tex]\(x^5\)[/tex], and [tex]\(x^2\)[/tex].
- The GCF of these exponents is [tex]\(x^2\)[/tex], as it is the lowest power of [tex]\(x\)[/tex] present in all terms.
3. Factor out the GCF:
- Now that we've determined the GCF to be [tex]\(-9x^2\)[/tex], we factor it out from each term in the expression:
[tex]\[
-45x^8 - 27x^5 - 18x^2 = -9x^2(5x^6 + 3x^3 + 2)
\][/tex]
4. Verify by distributing:
- To ensure the factoring is correct, distribute [tex]\(-9x^2\)[/tex]:
- Calculate [tex]\(-9x^2 \times 5x^6 = -45x^8\)[/tex]
- Calculate [tex]\(-9x^2 \times 3x^3 = -27x^5\)[/tex]
- Calculate [tex]\(-9x^2 \times 2 = -18x^2\)[/tex]
- These results match the original expression, confirming the factoring is accurate.
Thus, the expression [tex]\(-45x^8 - 27x^5 - 18x^2\)[/tex] factors to [tex]\(-9x^2(5x^6 + 3x^3 + 2)\)[/tex].
The correct answer is A: [tex]\(-9x^2(5x^6 + 3x^3 + 2)\)[/tex].
