To find the greatest common factor (GCF) of the terms in the polynomial [tex]\(12x^4 + 27x^3 + 6x^2\)[/tex], follow these steps:
1. Identify the coefficients and powers of [tex]\(x\)[/tex]:
- The coefficients in the polynomial are 12, 27, and 6.
- The powers of [tex]\(x\)[/tex] are 4, 3, and 2.
2. Find the greatest common divisor (GCD) of the coefficients:
- The coefficients are 12, 27, and 6. The GCD of these numbers needs to be determined.
- The GCD of 12 and 27 is 3.
- Now, find the GCD of 3 and 6, which is also 3.
- Therefore, the GCD of 12, 27, and 6 is 3.
3. Find the smallest power of [tex]\(x\)[/tex]:
- The powers of [tex]\(x\)[/tex] in the polynomial are 4, 3, and 2.
- The smallest power among these is 2.
4. Combine the GCD of the coefficients with the smallest power of [tex]\(x\)[/tex]:
- The GCD of the coefficients is 3.
- The smallest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].
Thus, the greatest common factor of the terms in the polynomial [tex]\(12x^4 + 27x^3 + 6x^2\)[/tex] is [tex]\(3x^2\)[/tex]. Therefore, the answer is [tex]\(3 x^2\)[/tex].
