Answer :
Let's factor the expression [tex]\(-45x^8 - 27x^5 - 18x^2\)[/tex] by identifying the greatest common factor (GCF).
### Step 1: Identify the GCF of the coefficients.
- For the coefficients [tex]\(-45\)[/tex], [tex]\(-27\)[/tex], and [tex]\(-18\)[/tex], we have:
- [tex]\(-45\)[/tex] can be factored as [tex]\(-1 \times 3^2 \times 5\)[/tex].
- [tex]\(-27\)[/tex] can be factored as [tex]\(-1 \times 3^3\)[/tex].
- [tex]\(-18\)[/tex] can be factored as [tex]\(-1 \times 2 \times 3^2\)[/tex].
The GCF of [tex]\(-45\)[/tex], [tex]\(-27\)[/tex], and [tex]\(-18\)[/tex] is [tex]\(-9\)[/tex].
### Step 2: Identify the GCF of the variable parts.
- For the variable parts [tex]\(x^8\)[/tex], [tex]\(x^5\)[/tex], and [tex]\(x^2\)[/tex], the common factor is the smallest power of [tex]\(x\)[/tex], which is [tex]\(x^2\)[/tex].
### Step 3: Combine the GCF of coefficients and variables.
So, the GCF of the entire expression is [tex]\(-9x^2\)[/tex].
### Step 4: Factor the GCF out of the expression.
Divide each term of the expression by the GCF [tex]\(-9x^2\)[/tex]:
- [tex]\(-45x^8 \div (-9x^2) = 5x^6\)[/tex]
- [tex]\(-27x^5 \div (-9x^2) = 3x^3\)[/tex]
- [tex]\(-18x^2 \div (-9x^2) = 2\)[/tex]
### Step 5: Write the factored expression.
Now, rewrite the expression using the GCF and the simplified terms:
[tex]\[
-45x^8 - 27x^5 - 18x^2 = -9x^2(5x^6 + 3x^3 + 2)
\][/tex]
Thus, the factored expression is [tex]\(-9x^2(5x^6 + 3x^3 + 2)\)[/tex].
The correct answer is A: [tex]\(-9x^2(5x^6 + 3x^3 + 2)\)[/tex].
### Step 1: Identify the GCF of the coefficients.
- For the coefficients [tex]\(-45\)[/tex], [tex]\(-27\)[/tex], and [tex]\(-18\)[/tex], we have:
- [tex]\(-45\)[/tex] can be factored as [tex]\(-1 \times 3^2 \times 5\)[/tex].
- [tex]\(-27\)[/tex] can be factored as [tex]\(-1 \times 3^3\)[/tex].
- [tex]\(-18\)[/tex] can be factored as [tex]\(-1 \times 2 \times 3^2\)[/tex].
The GCF of [tex]\(-45\)[/tex], [tex]\(-27\)[/tex], and [tex]\(-18\)[/tex] is [tex]\(-9\)[/tex].
### Step 2: Identify the GCF of the variable parts.
- For the variable parts [tex]\(x^8\)[/tex], [tex]\(x^5\)[/tex], and [tex]\(x^2\)[/tex], the common factor is the smallest power of [tex]\(x\)[/tex], which is [tex]\(x^2\)[/tex].
### Step 3: Combine the GCF of coefficients and variables.
So, the GCF of the entire expression is [tex]\(-9x^2\)[/tex].
### Step 4: Factor the GCF out of the expression.
Divide each term of the expression by the GCF [tex]\(-9x^2\)[/tex]:
- [tex]\(-45x^8 \div (-9x^2) = 5x^6\)[/tex]
- [tex]\(-27x^5 \div (-9x^2) = 3x^3\)[/tex]
- [tex]\(-18x^2 \div (-9x^2) = 2\)[/tex]
### Step 5: Write the factored expression.
Now, rewrite the expression using the GCF and the simplified terms:
[tex]\[
-45x^8 - 27x^5 - 18x^2 = -9x^2(5x^6 + 3x^3 + 2)
\][/tex]
Thus, the factored expression is [tex]\(-9x^2(5x^6 + 3x^3 + 2)\)[/tex].
The correct answer is A: [tex]\(-9x^2(5x^6 + 3x^3 + 2)\)[/tex].
