Answer :
To find the greatest common factor (GCF) for the list of terms [tex]\(30x^4\)[/tex], [tex]\(110x^5\)[/tex], and [tex]\(60x^8\)[/tex], follow these steps:
1. Find the GCF of the coefficients:
The coefficients of the terms are 30, 110, and 60. Start by finding the greatest common divisor (GCD) of these numbers.
- First, find the prime factorization of each coefficient:
- [tex]\(30 = 2 \times 3 \times 5\)[/tex]
- [tex]\(110 = 2 \times 5 \times 11\)[/tex]
- [tex]\(60 = 2 \times 2 \times 3 \times 5\)[/tex]
- Identify the common factors among these prime factorizations. The common prime factors are 2 and 5.
- Multiply the common prime factors to find the GCF:
- The GCF of 30, 110, and 60 is [tex]\(2 \times 5 = 10\)[/tex].
2. Find the GCF of the variables:
The terms have variables with different powers of [tex]\(x\)[/tex]: [tex]\(x^4\)[/tex], [tex]\(x^5\)[/tex], and [tex]\(x^8\)[/tex]. To find the GCF of the variable part, take the smallest exponent of [tex]\(x\)[/tex] present in these terms.
- The exponents of [tex]\(x\)[/tex] are 4, 5, and 8.
- The smallest exponent is 4.
3. Combine the results:
Combine the results from the coefficients and the smallest power of [tex]\(x\)[/tex] to determine the GCF of the terms.
- The GCF of the coefficients is 10.
- The smallest exponent of [tex]\(x\)[/tex] is 4.
Therefore, the greatest common factor of the terms [tex]\(30x^4\)[/tex], [tex]\(110x^5\)[/tex], and [tex]\(60x^8\)[/tex] is [tex]\(10x^4\)[/tex].
1. Find the GCF of the coefficients:
The coefficients of the terms are 30, 110, and 60. Start by finding the greatest common divisor (GCD) of these numbers.
- First, find the prime factorization of each coefficient:
- [tex]\(30 = 2 \times 3 \times 5\)[/tex]
- [tex]\(110 = 2 \times 5 \times 11\)[/tex]
- [tex]\(60 = 2 \times 2 \times 3 \times 5\)[/tex]
- Identify the common factors among these prime factorizations. The common prime factors are 2 and 5.
- Multiply the common prime factors to find the GCF:
- The GCF of 30, 110, and 60 is [tex]\(2 \times 5 = 10\)[/tex].
2. Find the GCF of the variables:
The terms have variables with different powers of [tex]\(x\)[/tex]: [tex]\(x^4\)[/tex], [tex]\(x^5\)[/tex], and [tex]\(x^8\)[/tex]. To find the GCF of the variable part, take the smallest exponent of [tex]\(x\)[/tex] present in these terms.
- The exponents of [tex]\(x\)[/tex] are 4, 5, and 8.
- The smallest exponent is 4.
3. Combine the results:
Combine the results from the coefficients and the smallest power of [tex]\(x\)[/tex] to determine the GCF of the terms.
- The GCF of the coefficients is 10.
- The smallest exponent of [tex]\(x\)[/tex] is 4.
Therefore, the greatest common factor of the terms [tex]\(30x^4\)[/tex], [tex]\(110x^5\)[/tex], and [tex]\(60x^8\)[/tex] is [tex]\(10x^4\)[/tex].
