Answer :
To find the greatest common factor (GCF) of the terms in the polynomial [tex]\(15x^4 + 6x^3 + 21x^2\)[/tex], follow these steps:
1. Identify the Coefficients and Exponents:
- Look at each term in the polynomial:
- The first term is [tex]\(15x^4\)[/tex], with a coefficient of 15 and an exponent of 4.
- The second term is [tex]\(6x^3\)[/tex], with a coefficient of 6 and an exponent of 3.
- The third term is [tex]\(21x^2\)[/tex], with a coefficient of 21 and an exponent of 2.
2. Find the GCF of the Coefficients:
- List the coefficients: 15, 6, and 21.
- Find the greatest common factor of these numbers:
- The factors of 15 are 1, 3, 5, 15.
- The factors of 6 are 1, 2, 3, 6.
- The factors of 21 are 1, 3, 7, 21.
- The greatest number that is a factor of all three coefficients is 3.
3. Find the GCF of the Variable Parts:
- Consider the variable parts: [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- Take the smallest power of [tex]\(x\)[/tex] present in all terms, which is [tex]\(x^2\)[/tex].
4. Combine the Results:
- The GCF of the coefficients is 3.
- The GCF of the variable parts is [tex]\(x^2\)[/tex].
- Therefore, the greatest common factor of the polynomial is [tex]\(3x^2\)[/tex].
The GCF of the terms in the polynomial [tex]\(15x^4 + 6x^3 + 21x^2\)[/tex] is [tex]\(3x^2\)[/tex].
1. Identify the Coefficients and Exponents:
- Look at each term in the polynomial:
- The first term is [tex]\(15x^4\)[/tex], with a coefficient of 15 and an exponent of 4.
- The second term is [tex]\(6x^3\)[/tex], with a coefficient of 6 and an exponent of 3.
- The third term is [tex]\(21x^2\)[/tex], with a coefficient of 21 and an exponent of 2.
2. Find the GCF of the Coefficients:
- List the coefficients: 15, 6, and 21.
- Find the greatest common factor of these numbers:
- The factors of 15 are 1, 3, 5, 15.
- The factors of 6 are 1, 2, 3, 6.
- The factors of 21 are 1, 3, 7, 21.
- The greatest number that is a factor of all three coefficients is 3.
3. Find the GCF of the Variable Parts:
- Consider the variable parts: [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- Take the smallest power of [tex]\(x\)[/tex] present in all terms, which is [tex]\(x^2\)[/tex].
4. Combine the Results:
- The GCF of the coefficients is 3.
- The GCF of the variable parts is [tex]\(x^2\)[/tex].
- Therefore, the greatest common factor of the polynomial is [tex]\(3x^2\)[/tex].
The GCF of the terms in the polynomial [tex]\(15x^4 + 6x^3 + 21x^2\)[/tex] is [tex]\(3x^2\)[/tex].
