Answer :
Sure! Let's factor the greatest common factor (GCF) out of the polynomial [tex]\(12x^7 + 28x^5 + 8x^4\)[/tex].
### Step-by-Step Solution
1. Identify the coefficients:
The coefficients of the polynomial are 12, 28, and 8.
2. Find the GCF of the coefficients:
- The factors of 12 are 1, 2, 3, 4, 6, and 12.
- The factors of 28 are 1, 2, 4, 7, 14, and 28.
- The factors of 8 are 1, 2, 4, and 8.
- The greatest common factor among these is 4.
3. Factor out the GCF from each term:
- Divide each coefficient by the GCF (which is 4), and factor it out.
- [tex]\(12 \div 4 = 3\)[/tex], so the first term becomes [tex]\(3x^7\)[/tex].
- [tex]\(28 \div 4 = 7\)[/tex], so the second term becomes [tex]\(7x^5\)[/tex].
- [tex]\(8 \div 4 = 2\)[/tex], so the third term becomes [tex]\(2x^4\)[/tex].
4. Write the factored form:
- Now that we have factored 4 out of each term, write it out as:
[tex]\[
4(3x^7 + 7x^5 + 2x^4)
\][/tex]
Thus, the polynomial [tex]\(12x^7 + 28x^5 + 8x^4\)[/tex] factored by its greatest common factor is [tex]\(4(3x^7 + 7x^5 + 2x^4)\)[/tex].
### Step-by-Step Solution
1. Identify the coefficients:
The coefficients of the polynomial are 12, 28, and 8.
2. Find the GCF of the coefficients:
- The factors of 12 are 1, 2, 3, 4, 6, and 12.
- The factors of 28 are 1, 2, 4, 7, 14, and 28.
- The factors of 8 are 1, 2, 4, and 8.
- The greatest common factor among these is 4.
3. Factor out the GCF from each term:
- Divide each coefficient by the GCF (which is 4), and factor it out.
- [tex]\(12 \div 4 = 3\)[/tex], so the first term becomes [tex]\(3x^7\)[/tex].
- [tex]\(28 \div 4 = 7\)[/tex], so the second term becomes [tex]\(7x^5\)[/tex].
- [tex]\(8 \div 4 = 2\)[/tex], so the third term becomes [tex]\(2x^4\)[/tex].
4. Write the factored form:
- Now that we have factored 4 out of each term, write it out as:
[tex]\[
4(3x^7 + 7x^5 + 2x^4)
\][/tex]
Thus, the polynomial [tex]\(12x^7 + 28x^5 + 8x^4\)[/tex] factored by its greatest common factor is [tex]\(4(3x^7 + 7x^5 + 2x^4)\)[/tex].
