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Answer :
Let's factor the polynomial [tex]\(21x^7 + 7x^5 - 21x^4\)[/tex] completely.
1. Identify the common factor:
Check each term:
- [tex]\(21x^7\)[/tex]
- [tex]\(7x^5\)[/tex]
- [tex]\(-21x^4\)[/tex]
We notice that each of these terms has a common factor of [tex]\(7x^4\)[/tex].
2. Factor out the common factor:
Factoring [tex]\(7x^4\)[/tex] from every term, we get:
[tex]\[
21x^7 + 7x^5 - 21x^4 = 7x^4(3x^3 + x - 3)
\][/tex]
3. Check for further factorization:
- The polynomial inside the parentheses is [tex]\(3x^3 + x - 3\)[/tex].
- We need to determine if [tex]\(3x^3 + x - 3\)[/tex] can be factored further.
After examining [tex]\(3x^3 + x - 3\)[/tex], we find that it cannot be factored further using integer coefficients.
Therefore, the completely factored form of the polynomial is:
[tex]\[
7x^4(3x^3 + x - 3)
\][/tex]
So, the correct choice is:
A. [tex]\(21x^7 + 7x^5 - 21x^4 = 7x^4(3x^3 + x - 3)\)[/tex] (Factor completely.)
1. Identify the common factor:
Check each term:
- [tex]\(21x^7\)[/tex]
- [tex]\(7x^5\)[/tex]
- [tex]\(-21x^4\)[/tex]
We notice that each of these terms has a common factor of [tex]\(7x^4\)[/tex].
2. Factor out the common factor:
Factoring [tex]\(7x^4\)[/tex] from every term, we get:
[tex]\[
21x^7 + 7x^5 - 21x^4 = 7x^4(3x^3 + x - 3)
\][/tex]
3. Check for further factorization:
- The polynomial inside the parentheses is [tex]\(3x^3 + x - 3\)[/tex].
- We need to determine if [tex]\(3x^3 + x - 3\)[/tex] can be factored further.
After examining [tex]\(3x^3 + x - 3\)[/tex], we find that it cannot be factored further using integer coefficients.
Therefore, the completely factored form of the polynomial is:
[tex]\[
7x^4(3x^3 + x - 3)
\][/tex]
So, the correct choice is:
A. [tex]\(21x^7 + 7x^5 - 21x^4 = 7x^4(3x^3 + x - 3)\)[/tex] (Factor completely.)
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