Answer :
To factor the expression [tex]-45x^8 - 27x^5 - 18x^2[/tex] using the greatest common factor (GCF), we first need to find the GCF of the coefficients and the variable terms.
Find the GCF of the coefficients:
The coefficients are [tex]-45[/tex], [tex]-27[/tex], and [tex]-18[/tex]. Let's find the GCF of these numbers.
- Prime factorization of 45 is [tex]3^2 \times 5[/tex].
- Prime factorization of 27 is [tex]3^3[/tex].
- Prime factorization of 18 is [tex]2 \times 3^2[/tex].
The GCF is [tex]3^2 = 9[/tex], since 9 is the highest power of the common prime factor 3 in the given numbers.
Find the GCF of the variable terms:
The variable terms are [tex]x^8[/tex], [tex]x^5[/tex], and [tex]x^2[/tex]. The smallest power of [tex]x[/tex] in these terms is [tex]x^2[/tex].
So, the GCF for the variable part is [tex]x^2[/tex].
Combine the GCFs:
Combine the GCFs of the coefficients and variable terms to get [tex]-9x^2[/tex] (note the negative sign from the leading coefficient).
Factor the original expression:
Divide each term of the expression by [tex]-9x^2[/tex] and factor it out:
[tex]\frac{-45x^8}{-9x^2} = 5x^6[/tex]
[tex]\frac{-27x^5}{-9x^2} = 3x^3[/tex]
[tex]\frac{-18x^2}{-9x^2} = 2[/tex]Therefore, the expression [tex]-45x^8 - 27x^5 - 18x^2[/tex] factors to:
[tex]-9x^2(5x^6 + 3x^3 + 2)[/tex]
Based on the given options, the correct factored expression is choice A: [tex]-9x^2(5x^6 + 3x^3 + 2)[/tex].
