Answer :
To find the greatest common factor (GCF) of the expressions [tex]\(70x^3\)[/tex], [tex]\(20x^4\)[/tex], and [tex]\(30x^5\)[/tex], we need to determine both the GCF of the coefficients and the exponents of [tex]\(x\)[/tex] separately.
### Step 1: Find the GCF of the Coefficients
The coefficients of the given terms are:
- 70
- 20
- 30
To find the GCF of these numbers, we identify the largest number that divides each of them without leaving a remainder.
1. List the factors:
- 70: [tex]\(1, 2, 5, 7, 10, 14, 35, 70\)[/tex]
- 20: [tex]\(1, 2, 4, 5, 10, 20\)[/tex]
- 30: [tex]\(1, 2, 3, 5, 6, 10, 15, 30\)[/tex]
2. Identify the common factors: [tex]\(1, 2, 5, 10\)[/tex]
3. The largest of these common factors is 10.
Therefore, the GCF of the coefficients is 10.
### Step 2: Find the GCF of the Exponents of [tex]\(x\)[/tex]
The exponents of [tex]\(x\)[/tex] in the given terms are:
- For [tex]\(70x^3\)[/tex], the exponent is 3.
- For [tex]\(20x^4\)[/tex], the exponent is 4.
- For [tex]\(30x^5\)[/tex], the exponent is 5.
To find the GCF of the exponents, we take the smallest exponent since all terms include [tex]\(x\)[/tex].
- The smallest exponent is 3.
### Conclusion
The greatest common factor of the expressions is given by combining the GCF of the coefficients and the smallest exponent of [tex]\(x\)[/tex]:
- GCF of the coefficients: 10
- GCF of the exponents: 3
Thus, the greatest common factor of [tex]\(70x^3\)[/tex], [tex]\(20x^4\)[/tex], and [tex]\(30x^5\)[/tex] is [tex]\(10x^3\)[/tex].
### Step 1: Find the GCF of the Coefficients
The coefficients of the given terms are:
- 70
- 20
- 30
To find the GCF of these numbers, we identify the largest number that divides each of them without leaving a remainder.
1. List the factors:
- 70: [tex]\(1, 2, 5, 7, 10, 14, 35, 70\)[/tex]
- 20: [tex]\(1, 2, 4, 5, 10, 20\)[/tex]
- 30: [tex]\(1, 2, 3, 5, 6, 10, 15, 30\)[/tex]
2. Identify the common factors: [tex]\(1, 2, 5, 10\)[/tex]
3. The largest of these common factors is 10.
Therefore, the GCF of the coefficients is 10.
### Step 2: Find the GCF of the Exponents of [tex]\(x\)[/tex]
The exponents of [tex]\(x\)[/tex] in the given terms are:
- For [tex]\(70x^3\)[/tex], the exponent is 3.
- For [tex]\(20x^4\)[/tex], the exponent is 4.
- For [tex]\(30x^5\)[/tex], the exponent is 5.
To find the GCF of the exponents, we take the smallest exponent since all terms include [tex]\(x\)[/tex].
- The smallest exponent is 3.
### Conclusion
The greatest common factor of the expressions is given by combining the GCF of the coefficients and the smallest exponent of [tex]\(x\)[/tex]:
- GCF of the coefficients: 10
- GCF of the exponents: 3
Thus, the greatest common factor of [tex]\(70x^3\)[/tex], [tex]\(20x^4\)[/tex], and [tex]\(30x^5\)[/tex] is [tex]\(10x^3\)[/tex].
