How do you divide #(2x^4 x^3  9x^2  x + 5)/ (5x^2  2) # using polynomial long division?
This really is long division but its layout is different to the conventional approach.
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The remainder is
Apply the polynomial long division formula.
Consequently,
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To divide (2x^4  x^3  9x^2  x + 5) by (5x^2  2) using polynomial long division, follow these steps:

Arrange the dividend and divisor in descending order of exponents. Dividend: 2x^4  x^3  9x^2  x + 5 Divisor: 5x^2  2

Divide the leading term of the dividend (2x^4) by the leading term of the divisor (5x^2), which gives (2x^4)/(5x^2) = (2/5)x^2.

Multiply the divisor (5x^2  2) by the result obtained in step 2, (2/5)x^2. This gives (2/5)x^2 * (5x^2  2) = (2/5)x^4  (4/5)x^2.

Subtract the result obtained in step 3 from the dividend. (2x^4  x^3  9x^2  x + 5)  ((2/5)x^4  (4/5)x^2) = (8/5)x^4  x^3  (1/5)x^2  x + 5.

Bring down the next term from the dividend, which is x. The new dividend becomes (8/5)x^4  x^3  (1/5)x^2  x + 5  x = (8/5)x^4  x^3  (1/5)x^2  2x + 5.

Repeat steps 25 until the degree of the new dividend is less than the degree of the divisor.

Divide the leading term of the new dividend ((8/5)x^4) by the leading term of the divisor (5x^2), which gives ((8/5)x^4)/(5x^2) = (8/25)x^2.

Multiply the divisor (5x^2  2) by the result obtained in step 7, (8/25)x^2. This gives (8/25)x^2 * (5x^2  2) = (8/25)x^4  (16/25)x^2.

Subtract the result obtained in step 8 from the new dividend. (8/5)x^4  x^3  (1/5)x^2  2x + 5  ((8/25)x^4  (16/25)x^2) = (32/25)x^4  x^3 + (9/25)x^2  2x + 5.

Bring down the next term from the new dividend, which is 2x. The new dividend becomes (32/25)x^4  x^3 + (9/25)x^2  2x + 5  (2x) = (32/25)x^4  x^3 + (9/25)x^2.

Repeat steps 710 until the degree of the new dividend is less than the degree of the divisor.

Divide the leading term of the new dividend ((32/25)x^4) by the leading term of the divisor (5x^2), which gives ((32/25)x^4)/(5x^2) = (32/125)x^2.

Multiply the divisor (5x^2  2) by the result obtained in step 12, (32/125)x^2. This gives (32/125)x^2 * (5x^2  2) = (32/125)x^4  (64/125)x^2.

Subtract the result obtained in step 13 from the new dividend. (32/25)x^4  x^3 + (9/25)x^2  ((32/125)x^4  (64/125)x^2) = (96/125)x^4  x^3 + (73/125)x^2.

The degree of the new dividend (96/125)x^4  x^3 + (73/125)x^2 is less than the degree of the divisor (5x^2  2), so the division is complete.
The quotient is (2/5)x^2 + (8/25)x^2 + (32/125)x^2 = (2/5)x^2 + (8/25)x^2 + (32/125)x^2 = (2/5)x^2 + (8/25)x^2 + (32/125)x^2.
The remainder is (96/125)x^4  x^3 + (73/125)x^2.
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To divide (2x^4  x^3  9x^2  x + 5) by (5x^2  2) using polynomial long division:
 Divide the leading term of the dividend (2x^4) by the leading term of the divisor (5x^2) to get (2x^4) / (5x^2) = (2/5)x^2.
 Multiply the entire divisor by (2/5)x^2 to get (2/5)x^4  (4/5)x^2.
 Subtract this result from the dividend to get the remainder: (2x^4  x^3  9x^2  x + 5)  ((2/5)x^4  (4/5)x^2) = (8/5)x^4  x^3  (1/5)x^2  x + 5.
 Repeat the process with the new polynomial (8/5)x^4  x^3  (1/5)x^2  x + 5.
Continue this process until the degree of the remainder is less than the degree of the divisor.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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