College

Factor out the greatest common factor in the expression:

[tex]\[ 9x^4 - 18x^3 + 27x^2 \][/tex]

[tex]\[ 9x^4 - 18x^3 + 27x^2 = \square \][/tex]

Answer :

To factor out the greatest common factor (GCF) from the expression [tex]\(9x^4 - 18x^3 + 27x^2\)[/tex], we can follow these steps:

1. Identify the coefficients and their greatest common factor:

- The coefficients of the terms are 9, -18, and 27.
- The greatest common factor of these coefficients is 9.

2. Identify the variable part and its greatest common factor:

- Look at the powers of [tex]\(x\)[/tex] in each term: [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- The smallest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex], so [tex]\(x^2\)[/tex] is the greatest common factor for the [tex]\(x\)[/tex] terms.

3. Combine the results:

- The overall GCF of the expression is [tex]\(9x^2\)[/tex].

4. Factor out the GCF from each term:

- Divide each term by [tex]\(9x^2\)[/tex] and factor it out from the expression:

- [tex]\(9x^4 \div 9x^2 = x^2\)[/tex]
- [tex]\(-18x^3 \div 9x^2 = -2x\)[/tex]
- [tex]\(27x^2 \div 9x^2 = 3\)[/tex]

5. Write the factored expression:

- After factoring out the GCF, the expression becomes:
[tex]\[
9x^4 - 18x^3 + 27x^2 = 9x^2(x^2 - 2x + 3)
\][/tex]

So, the greatest common factor of the given expression [tex]\(9x^4 - 18x^3 + 27x^2\)[/tex] is successfully factored out, leaving us with:
[tex]\[ 9x^2(x^2 - 2x + 3) \][/tex]