Answer :
Sure, I'd be happy to help! Let's determine the greatest common factor (GCF) of the terms [tex]\(3x^4\)[/tex], [tex]\(15x^3\)[/tex], and [tex]\(21x^2\)[/tex].
### Step-by-Step Solution:
1. Factor the coefficients:
- The coefficients are 3, 15, and 21.
- Factor these numbers into primes:
- 3 is [tex]\(3 = 3\)[/tex]
- 15 is [tex]\(15 = 3 \times 5\)[/tex]
- 21 is [tex]\(21 = 3 \times 7\)[/tex]
2. Find the GCF of the coefficients:
- The common prime factor among 3, 15, and 21 is 3.
- So, the GCF of the coefficients is 3.
3. Determine the common powers of [tex]\(x\)[/tex]:
- The exponents of [tex]\(x\)[/tex] are 4, 3, and 2.
- The smallest exponent in these terms is 2.
4. Combine the results:
- The GCF of the coefficients is 3.
- The GCF of the powers of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].
Putting it all together, the greatest common factor (GCF) of [tex]\(3x^4\)[/tex], [tex]\(15x^3\)[/tex], and [tex]\(21x^2\)[/tex] is:
[tex]\[3x^2\][/tex]
So, the greatest common factor (GCF) is [tex]\(\boxed{3x^2}\)[/tex].
### Step-by-Step Solution:
1. Factor the coefficients:
- The coefficients are 3, 15, and 21.
- Factor these numbers into primes:
- 3 is [tex]\(3 = 3\)[/tex]
- 15 is [tex]\(15 = 3 \times 5\)[/tex]
- 21 is [tex]\(21 = 3 \times 7\)[/tex]
2. Find the GCF of the coefficients:
- The common prime factor among 3, 15, and 21 is 3.
- So, the GCF of the coefficients is 3.
3. Determine the common powers of [tex]\(x\)[/tex]:
- The exponents of [tex]\(x\)[/tex] are 4, 3, and 2.
- The smallest exponent in these terms is 2.
4. Combine the results:
- The GCF of the coefficients is 3.
- The GCF of the powers of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].
Putting it all together, the greatest common factor (GCF) of [tex]\(3x^4\)[/tex], [tex]\(15x^3\)[/tex], and [tex]\(21x^2\)[/tex] is:
[tex]\[3x^2\][/tex]
So, the greatest common factor (GCF) is [tex]\(\boxed{3x^2}\)[/tex].
