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]
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]
