Answer :
To find the greatest common factor (GCF) of the expression [tex]\(25x^4 + 10x^3 + 15x^2\)[/tex], follow these steps:
1. Identify the coefficients and the variable parts separately:
- The coefficients are [tex]\(25\)[/tex], [tex]\(10\)[/tex], and [tex]\(15\)[/tex].
- The variable parts are [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
2. Find the GCF of the coefficients:
- List the factors for each coefficient:
- Factors of [tex]\(25\)[/tex] are [tex]\(1, 5, 25\)[/tex].
- Factors of [tex]\(10\)[/tex] are [tex]\(1, 2, 5, 10\)[/tex].
- Factors of [tex]\(15\)[/tex] are [tex]\(1, 3, 5, 15\)[/tex].
- The common factors of these coefficients are [tex]\(1\)[/tex] and [tex]\(5\)[/tex].
- The greatest common factor of [tex]\(25\)[/tex], [tex]\(10\)[/tex], and [tex]\(15\)[/tex] is [tex]\(5\)[/tex].
3. Find the GCF of the variable parts:
- The variable parts are [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- The greatest common factor for these variable parts will be the lowest power of [tex]\(x\)[/tex] common to all, which is [tex]\(x^2\)[/tex].
4. Combine the GCFs of both parts:
- The GCF of the coefficients is [tex]\(5\)[/tex].
- The GCF of the variable parts is [tex]\(x^2\)[/tex].
Therefore, the greatest common factor (GCF) of the expression [tex]\(25x^4 + 10x^3 + 15x^2\)[/tex] is:
[tex]\[ 5x^2 \][/tex]
So, the final answer is [tex]\(5x^2\)[/tex].
1. Identify the coefficients and the variable parts separately:
- The coefficients are [tex]\(25\)[/tex], [tex]\(10\)[/tex], and [tex]\(15\)[/tex].
- The variable parts are [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
2. Find the GCF of the coefficients:
- List the factors for each coefficient:
- Factors of [tex]\(25\)[/tex] are [tex]\(1, 5, 25\)[/tex].
- Factors of [tex]\(10\)[/tex] are [tex]\(1, 2, 5, 10\)[/tex].
- Factors of [tex]\(15\)[/tex] are [tex]\(1, 3, 5, 15\)[/tex].
- The common factors of these coefficients are [tex]\(1\)[/tex] and [tex]\(5\)[/tex].
- The greatest common factor of [tex]\(25\)[/tex], [tex]\(10\)[/tex], and [tex]\(15\)[/tex] is [tex]\(5\)[/tex].
3. Find the GCF of the variable parts:
- The variable parts are [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- The greatest common factor for these variable parts will be the lowest power of [tex]\(x\)[/tex] common to all, which is [tex]\(x^2\)[/tex].
4. Combine the GCFs of both parts:
- The GCF of the coefficients is [tex]\(5\)[/tex].
- The GCF of the variable parts is [tex]\(x^2\)[/tex].
Therefore, the greatest common factor (GCF) of the expression [tex]\(25x^4 + 10x^3 + 15x^2\)[/tex] is:
[tex]\[ 5x^2 \][/tex]
So, the final answer is [tex]\(5x^2\)[/tex].
