Answer :
To factor out the greatest common factor (GCF) from the polynomial [tex]\(7x^6 - 21x^5 + 21x^4\)[/tex], follow these steps:
1. Identify the GCF of the coefficients:
- Look at the coefficients of each term in the polynomial: 7, 21, and 21.
- The GCF of 7, 21, and 21 is 7 since 7 is the largest number that divides all three coefficients without leaving a remainder.
2. Factor out the GCF from each term:
- Divide each coefficient by the GCF (7), and then factor 7 out from the polynomial.
- [tex]\(7x^6 ÷ 7 = x^6\)[/tex]
- [tex]\(-21x^5 ÷ 7 = -3x^5\)[/tex]
- [tex]\(21x^4 ÷ 7 = 3x^4\)[/tex]
3. Rewrite the polynomial with the GCF factored out:
- After factoring out the GCF of 7, the polynomial becomes:
[tex]\[
7(x^6 - 3x^5 + 3x^4)
\][/tex]
So, the greatest common factor of the polynomial is 7, and the polynomial can be expressed as [tex]\(7(x^6 - 3x^5 + 3x^4)\)[/tex]. This means the terms inside the parentheses represent the simplified expression of the original polynomial after factoring out the GCF.
1. Identify the GCF of the coefficients:
- Look at the coefficients of each term in the polynomial: 7, 21, and 21.
- The GCF of 7, 21, and 21 is 7 since 7 is the largest number that divides all three coefficients without leaving a remainder.
2. Factor out the GCF from each term:
- Divide each coefficient by the GCF (7), and then factor 7 out from the polynomial.
- [tex]\(7x^6 ÷ 7 = x^6\)[/tex]
- [tex]\(-21x^5 ÷ 7 = -3x^5\)[/tex]
- [tex]\(21x^4 ÷ 7 = 3x^4\)[/tex]
3. Rewrite the polynomial with the GCF factored out:
- After factoring out the GCF of 7, the polynomial becomes:
[tex]\[
7(x^6 - 3x^5 + 3x^4)
\][/tex]
So, the greatest common factor of the polynomial is 7, and the polynomial can be expressed as [tex]\(7(x^6 - 3x^5 + 3x^4)\)[/tex]. This means the terms inside the parentheses represent the simplified expression of the original polynomial after factoring out the GCF.
