High School

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Factor the expression using the greatest common factor.

[tex]-45 x^8 - 27 x^5 - 18 x^2[/tex]

A. [tex]-9 x^2(5 x^6 + 3 x^3 + 2)[/tex]
B. [tex]-9 x^2(5 x^6 + 2 x^3 + 3)[/tex]
C. [tex]-9(5 x^8 + 2 x^5 + 3)[/tex]
D. [tex]-9(5 x^8 + 3 x^5 + 2 x^2)[/tex]

Answer :

To factor the expression [tex]\(-45x^8 - 27x^5 - 18x^2\)[/tex] using the greatest common factor (GCF), follow these steps:

1. Identify the GCF for the coefficients:
- The coefficients are [tex]\(-45\)[/tex], [tex]\(-27\)[/tex], and [tex]\(-18\)[/tex].
- Find the GCF of these numbers. The GCF of [tex]\(-45\)[/tex], [tex]\(-27\)[/tex], and [tex]\(-18\)[/tex] is [tex]\(-9\)[/tex].

2. Identify the GCF for the variable terms:
- The powers of [tex]\(x\)[/tex] 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].
- Therefore, the GCF for the variable terms is [tex]\(x^2\)[/tex].

3. Factor out the GCF:
- The GCF for the entire expression is [tex]\(-9x^2\)[/tex].
- Divide each term in the expression by [tex]\(-9x^2\)[/tex] and place it outside the parentheses:
- [tex]\(-45x^8 ÷ -9x^2 = 5x^6\)[/tex]
- [tex]\(-27x^5 ÷ -9x^2 = 3x^3\)[/tex]
- [tex]\(-18x^2 ÷ -9x^2 = 2\)[/tex]

4. Write the factored expression:
- Combine the results into the factored form using the GCF:
[tex]\[ -9x^2(5x^6 + 3x^3 + 2) \][/tex]

Therefore, the expression [tex]\(-45x^8 - 27x^5 - 18x^2\)[/tex] is factored as [tex]\(-9x^2(5x^6 + 3x^3 + 2)\)[/tex]. This corresponds to option A.

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