Answer :
To find the Greatest Common Factor (GCF) of the polynomial [tex]\(9x^5 + 21x^4 - 45x^3\)[/tex], we proceed in the following steps:
1. Identify the Coefficients: Look at the numerical coefficients of each term in the polynomial. They are 9, 21, and -45.
2. Find the GCF of the Coefficients:
- The GCF is the largest number that divides each of the coefficients without leaving a remainder.
- For the coefficients 9, 21, and -45, first, we look at the prime factorization:
- 9 can be factored as [tex]\(3 \times 3\)[/tex].
- 21 can be factored as [tex]\(3 \times 7\)[/tex].
- -45 can be factored as [tex]\(-1 \times 3 \times 3 \times 5\)[/tex].
- The common factor among these numbers is 3.
3. Check the Variable Terms:
- Each term in the polynomial contains the variable [tex]\(x\)[/tex].
- Identify the lowest power of [tex]\(x\)[/tex] present in all terms. The minimum power of [tex]\(x\)[/tex] in the polynomial is [tex]\(x^3\)[/tex].
4. Combine the GCFs:
- Combine the GCF of the coefficients (3) with the smallest power of [tex]\(x\)[/tex] present (which is [tex]\(x^3\)[/tex]).
- Thus, the GCF of the whole expression [tex]\(9x^5 + 21x^4 - 45x^3\)[/tex] is [tex]\(3x^3\)[/tex].
5. Factoring the Polynomial:
- Factor out the GCF [tex]\(3x^3\)[/tex] from the polynomial:
[tex]\[
9x^5 + 21x^4 - 45x^3 = 3x^3(3x^2 + 7x - 15)
\][/tex]
Therefore, the Greatest Common Factor (GCF) of the polynomial [tex]\(9x^5 + 21x^4 - 45x^3\)[/tex] is [tex]\(3x^3\)[/tex], and the polynomial can be factored as [tex]\(3x^3(3x^2 + 7x - 15)\)[/tex].
1. Identify the Coefficients: Look at the numerical coefficients of each term in the polynomial. They are 9, 21, and -45.
2. Find the GCF of the Coefficients:
- The GCF is the largest number that divides each of the coefficients without leaving a remainder.
- For the coefficients 9, 21, and -45, first, we look at the prime factorization:
- 9 can be factored as [tex]\(3 \times 3\)[/tex].
- 21 can be factored as [tex]\(3 \times 7\)[/tex].
- -45 can be factored as [tex]\(-1 \times 3 \times 3 \times 5\)[/tex].
- The common factor among these numbers is 3.
3. Check the Variable Terms:
- Each term in the polynomial contains the variable [tex]\(x\)[/tex].
- Identify the lowest power of [tex]\(x\)[/tex] present in all terms. The minimum power of [tex]\(x\)[/tex] in the polynomial is [tex]\(x^3\)[/tex].
4. Combine the GCFs:
- Combine the GCF of the coefficients (3) with the smallest power of [tex]\(x\)[/tex] present (which is [tex]\(x^3\)[/tex]).
- Thus, the GCF of the whole expression [tex]\(9x^5 + 21x^4 - 45x^3\)[/tex] is [tex]\(3x^3\)[/tex].
5. Factoring the Polynomial:
- Factor out the GCF [tex]\(3x^3\)[/tex] from the polynomial:
[tex]\[
9x^5 + 21x^4 - 45x^3 = 3x^3(3x^2 + 7x - 15)
\][/tex]
Therefore, the Greatest Common Factor (GCF) of the polynomial [tex]\(9x^5 + 21x^4 - 45x^3\)[/tex] is [tex]\(3x^3\)[/tex], and the polynomial can be factored as [tex]\(3x^3(3x^2 + 7x - 15)\)[/tex].
