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To find the Greatest Common Factor (GCF) of each of the following sets of expressions, we will examine the coefficients and variables involved for each set separately. Let's go through them step-by-step:
1. Finding the GCF of 12, 15, and 18:
- First, we list the factors of each number.
- 12: 1, 2, 3, 4, 6, 12
- 15: 1, 3, 5, 15
- 18: 1, 2, 3, 6, 9, 18
- We then identify the common factors: 1 and 3.
- The greatest common factor among these numbers is 3.
Therefore, the GCF of 12, 15, and 18 is 3.
2. Finding the GCF of [tex]\(x^4, x^3, x^2\)[/tex]:
- For variables, the GCF is determined by the lowest power of the variable common in all terms.
- The powers of x are 4, 3, and 2 respectively.
- The smallest power of x common to all terms is [tex]\(x^2\)[/tex].
Therefore, the GCF of [tex]\(x^4, x^3, x^2\)[/tex] is [tex]\(x^2\)[/tex].
3. Finding the GCF of [tex]\(5a^3, 10a^2, 15a\)[/tex]:
- First, we list the coefficients: 5, 10, and 15.
- The GCF of these coefficients (5, 10, and 15) is found by identifying the highest number that divides all of them evenly, which is 5.
- Next, considering the variable [tex]\(a\)[/tex]:
- [tex]\(a^3, a^2,\)[/tex] and [tex]\(a\)[/tex] all have a common factor of [tex]\(a\)[/tex], and the smallest power is [tex]\(a\)[/tex].
Therefore, the GCF of [tex]\(5a^3, 10a^2, 15a\)[/tex] is 5a.
4. Finding the GCF of 12a and 18abc:
- First, we consider the coefficients: 12 and 18.
- The GCF of 12 and 18 is 6.
- Next, for the variables [tex]\(a\)[/tex] and [tex]\(abc\)[/tex]:
- The common variable factor is [tex]\(a\)[/tex].
Therefore, the GCF of 12a and 18abc is 6a.
5. Finding the GCF of [tex]\(6a^2b^2\)[/tex] and [tex]\(3ab^2\)[/tex]:
- Consider the coefficients first: 6 and 3.
- The GCF of 6 and 3 is 3.
- Next, for the variables:
- The common factor is [tex]\(a\)[/tex] with the smallest power being [tex]\(a\)[/tex].
- The common factor for [tex]\(b\)[/tex] is [tex]\(b^2\)[/tex].
Therefore, the GCF of [tex]\(6a^2b^2\)[/tex] and [tex]\(3ab^2\)[/tex] is 3ab^2.
Summarizing:
1. [tex]\(12, 15, 18 \quad\)[/tex] GCF = 3
2. [tex]\(x^4, x^3, x^2 \quad\)[/tex] GCF = [tex]\(x^2\)[/tex]
3. [tex]\(5a^3, 10a^2, 15a \quad\)[/tex] GCF = 5a
4. [tex]\(12a\)[/tex] and [tex]\(18abc\)[/tex] GCF = 6a
5. [tex]\(6a^2b^2\)[/tex] and [tex]\(3ab^2\)[/tex] GCF = 3ab^2
1. Finding the GCF of 12, 15, and 18:
- First, we list the factors of each number.
- 12: 1, 2, 3, 4, 6, 12
- 15: 1, 3, 5, 15
- 18: 1, 2, 3, 6, 9, 18
- We then identify the common factors: 1 and 3.
- The greatest common factor among these numbers is 3.
Therefore, the GCF of 12, 15, and 18 is 3.
2. Finding the GCF of [tex]\(x^4, x^3, x^2\)[/tex]:
- For variables, the GCF is determined by the lowest power of the variable common in all terms.
- The powers of x are 4, 3, and 2 respectively.
- The smallest power of x common to all terms is [tex]\(x^2\)[/tex].
Therefore, the GCF of [tex]\(x^4, x^3, x^2\)[/tex] is [tex]\(x^2\)[/tex].
3. Finding the GCF of [tex]\(5a^3, 10a^2, 15a\)[/tex]:
- First, we list the coefficients: 5, 10, and 15.
- The GCF of these coefficients (5, 10, and 15) is found by identifying the highest number that divides all of them evenly, which is 5.
- Next, considering the variable [tex]\(a\)[/tex]:
- [tex]\(a^3, a^2,\)[/tex] and [tex]\(a\)[/tex] all have a common factor of [tex]\(a\)[/tex], and the smallest power is [tex]\(a\)[/tex].
Therefore, the GCF of [tex]\(5a^3, 10a^2, 15a\)[/tex] is 5a.
4. Finding the GCF of 12a and 18abc:
- First, we consider the coefficients: 12 and 18.
- The GCF of 12 and 18 is 6.
- Next, for the variables [tex]\(a\)[/tex] and [tex]\(abc\)[/tex]:
- The common variable factor is [tex]\(a\)[/tex].
Therefore, the GCF of 12a and 18abc is 6a.
5. Finding the GCF of [tex]\(6a^2b^2\)[/tex] and [tex]\(3ab^2\)[/tex]:
- Consider the coefficients first: 6 and 3.
- The GCF of 6 and 3 is 3.
- Next, for the variables:
- The common factor is [tex]\(a\)[/tex] with the smallest power being [tex]\(a\)[/tex].
- The common factor for [tex]\(b\)[/tex] is [tex]\(b^2\)[/tex].
Therefore, the GCF of [tex]\(6a^2b^2\)[/tex] and [tex]\(3ab^2\)[/tex] is 3ab^2.
Summarizing:
1. [tex]\(12, 15, 18 \quad\)[/tex] GCF = 3
2. [tex]\(x^4, x^3, x^2 \quad\)[/tex] GCF = [tex]\(x^2\)[/tex]
3. [tex]\(5a^3, 10a^2, 15a \quad\)[/tex] GCF = 5a
4. [tex]\(12a\)[/tex] and [tex]\(18abc\)[/tex] GCF = 6a
5. [tex]\(6a^2b^2\)[/tex] and [tex]\(3ab^2\)[/tex] GCF = 3ab^2
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