# How do you simplify #(25a^4 + 19a^2 - a - 78)/(5a^2 + 6a - 9)#?

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To simplify the expression (25a^4 + 19a^2 - a - 78)/(5a^2 + 6a - 9), you can use polynomial long division or synthetic division. However, in this case, polynomial long division is more suitable. Here are the steps:

- Divide the first term of the numerator (25a^4) by the first term of the denominator (5a^2) to get 5a^2.
- Multiply the entire denominator (5a^2 + 6a - 9) by the result from step 1 (5a^2) to get 25a^4 + 30a^3 - 45a^2.
- Subtract the result from step 2 from the numerator (25a^4 + 19a^2 - a - 78) to get 30a^3 + 64a^2 - a - 78.
- Repeat steps 1-3 with the new expression (30a^3 + 64a^2 - a - 78) and the denominator (5a^2 + 6a - 9).
- Divide the first term of the new expression (30a^3) by the first term of the denominator (5a^2) to get 6a.
- Multiply the entire denominator (5a^2 + 6a - 9) by the result from step 4 (6a) to get 30a^3 + 36a^2 - 54a.
- Subtract the result from step 5 from the new expression (30a^3 + 64a^2 - a - 78) to get 28a^2 + 53a - 78.

- Repeat steps 4-5 until you can no longer divide.
- Divide the first term of the new expression (28a^2) by the first term of the denominator (5a^2) to get 5.6.
- Multiply the entire denominator (5a^2 + 6a - 9) by the result from step 6 (5.6) to get 28a^2 + 33.6a - 50.4.
- Subtract the result from step 7 from the new expression (28a^2 + 53a - 78) to get 19.4a - 27.6.

The simplified form of the expression is 5a^2 + 6a - 9 with a remainder of 19.4a - 27.6.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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