To determine the correct factorizations of the polynomial [tex]\( 27x^6 - 45x^4 + 63x^2 \)[/tex], let’s examine each provided form and verify their correctness.
1. Factor form: [tex]\( 9x^2\left( 3x^4 - 5x^2 + 7 \right) \)[/tex]
- To verify, we can expand this expression:
[tex]\[
9x^2 \cdot (3x^4 - 5x^2 + 7) = 9x^2 \cdot 3x^4 + 9x^2 \cdot (-5x^2) + 9x^2 \cdot 7
\][/tex]
Simplifying each term individually:
[tex]\[
9x^2 \cdot 3x^4 = 27x^6
\][/tex]
[tex]\[
9x^2 \cdot (-5x^2) = -45x^4
\][/tex]
[tex]\[
9x^2 \cdot 7 = 63x^2
\][/tex]
Therefore, the expanded form is:
[tex]\[
27x^6 - 45x^4 + 63x^2
\][/tex]
This matches the given polynomial, so this factorization is correct.
2. Factor form: [tex]\( 9x^2 \left( 3x^4 - 5x^2 + 7 \right) \)[/tex]
- This is identical to the first factor form we verified and is also correct for the reasons given above.
3. Factor form: [tex]\( 9x^2 \cdot 3x^4 - 9x^2 \cdot 5x^2 + 9x^2 \cdot 7 \)[/tex]
- To verify, we can expand this expression:
[tex]\[
9x^2 \cdot 3x^4 - 9x^2 \cdot 5x^2 + 9x^2 \cdot 7 = 27x^6 - 45x^4 + 63x^2
\][/tex]
This matches the given polynomial directly. Thus, this factorization is also correct.
4. Factor form: [tex]\( 3x^2 \left( 9x^4 - 15x^2 + 21 \right) \)[/tex]
- To verify, we can expand this expression:
[tex]\[
3x^2 \cdot (9x^4 - 15x^2 + 21) = 3x^4 \cdot 9x^4 + 3x^2 \cdot (-15x^2) + 3x^2 \cdot 21
\][/tex]
Simplifying each term individually:
[tex]\[
3x^2 \cdot 9x^4 = 27x^6
\][/tex]
[tex]\[
3x^2 \cdot (-15x^2) = -45x^4
\][/tex]
[tex]\[
3x^2 \cdot 21 = 63x^2
\][/tex]
Therefore, the expanded form is:
[tex]\[
27x^6 - 45x^4 + 63x^2
\][/tex]
This matches the given polynomial, so this factorization is also correct.
Conclusion:
All provided factor forms match the given polynomial when expanded. Thus, the correct factorizations are:
[tex]\[
9x^2 (3x^4 - 5x^2 + 7), \quad 9x^2 (3x^4 - 5x^2 + 7), \quad 9x^2 \cdot 3x^4 - 9x^2 \cdot 5x^2 + 9x^2 \cdot 7, \quad 3x^2 (9x^4 - 15x^2 + 21)
\][/tex]
