To divide the polynomial \( 23a^3 \) by \( -a \), we perform the division:
\[ \frac{23a^3}{-a} = -23a^2 \]
So, the result of dividing \( 23a^3 \) by \( -a \) is \( -23a^2 \).
Now, let's analyze the polynomial identities given in the options. We can rewrite \( -a^3 \) as follows:
\[ -a^3 = (2 - a)(\text{some polynomial}) \]
To find that "some polynomial," we can use polynomial long division or synthetic division, but since the question suggests establishing an identity that relates to \( -a^3 \) and its factors, we can check which provided option correctly satisfies this.
Let's check the options given and see if they approximate the factorization of \( -a^3 \).
Looking at the options:
- Option 1: \( -a^3 = (2 - a)(22 + ax + a^2) \)
- Option 2: \( -a^3 = (2 - a)(22 + ax - a^2) \)
- Option 3: \( -a^3 = (r - a)(2^2 - ax + a^2) \) \( [\text{r appears problematic}] \)
- Option 4: \( -a^3 = (2 - a)(22 - ax - a^2) \)
Since polynomial identities provide equal results when you substitute for \( a \), we need to check which polynomial identity holds by expansion.
Calculating Product of Each Option:
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Option 1: \[ (2 - a)(22 + ax + a^2) = 2 \cdot 22 + 2 \cdot ax + 2 \cdot a^2 - 22a - ax^2 - a^3 \] Resulting in some terms that will not equal \( -a^3 \).
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Option 2: \[ (2 - a)(22 + ax - a^2) = 44 + 2ax - 2a^2 - 22a - ax^2 + a^3 \] Again does not equal \( -a^3 \).
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Option 3: This doesn’t make sense because of the \( r \) without specification.
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Option 4: \[ (2 - a)(22 - ax - a^2) = 44 - 2ax - 2a^2 - 22a + ax^2 + a^3 \] This one can be reorganized and simplified but again does not simply give identity needed.
Conclusively:
None of the identified polynomial identities precisely yield \( -a^3 \) when calculated in current form. Reviewing errors assumed in variable choices and misinterpretations become final checks.
The corrected answer should use separation rules, leading to identification clarity among options. If struggling for direct verification, understand polynomial division produces coefficients collected from separate factor functions.
As per verification of method revisions, none of the specified identities accurately represent the polynomial \( 23a^3 \) against the conditions from the options provided. Reanalysis might encourage further reconstruction methodically through polynomial deduction strategies.
Should you require further assistance in deeper factor checks or corrections, don't hesitate to mention specific polynomial identities lined against original constructs.
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