How do you divide #(x^3 + x^2  x  1)/(x  1) # using polynomial long division?
Long division of polynomials is similar to long division of numbers.
See explanation...
Here's an animation of the process:
Write the dividend under the bar and the divisor to the left. Each is written in descending order of powers of
Choose the first term of the quotient to cause leading terms to match. In our example, we choose
Write the product of this term and the divisor below the dividend and subtract to give a remainder (
Bring down the next term (
Choose the next term (
Stop when there is nothing more to bring down from the dividend and the running remainder has lower degree than the divisor.
In our example, the division is exact. We are left with no remainder.
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To divide (x^3 + x^2  x  1) by (x  1) using polynomial long division, follow these steps:

Write the dividend (x^3 + x^2  x  1) and the divisor (x  1) in long division format.

Divide the first term of the dividend (x^3) by the first term of the divisor (x). The result is x^2.

Multiply the divisor (x  1) by the result obtained in step 2 (x^2). The product is x^3  x^2.

Subtract the product obtained in step 3 from the dividend. (x^3 + x^2  x  1)  (x^3  x^2) = 2x^2  x  1.

Bring down the next term from the dividend, which is x. The new dividend is 2x^2  x  1.

Divide the first term of the new dividend (2x^2) by the first term of the divisor (x). The result is 2x.

Multiply the divisor (x  1) by the result obtained in step 6 (2x). The product is 2x^2  2x.

Subtract the product obtained in step 7 from the new dividend. (2x^2  x  1)  (2x^2  2x) = x  1.

Bring down the next term from the new dividend, which is 1. The new dividend is x  1.

Divide the first term of the new dividend (x) by the first term of the divisor (x). The result is 1.

Multiply the divisor (x  1) by the result obtained in step 10 (1). The product is x  1.

Subtract the product obtained in step 11 from the new dividend. (x  1)  (x  1) = 0.

There is no remainder, and the quotient is x^2 + 2x + 1.
Therefore, (x^3 + x^2  x  1)/(x  1) = x^2 + 2x + 1.
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To divide ( \frac{x^3 + x^2  x  1}{x  1} ) using polynomial long division:

Divide the first term of the numerator by the first term of the denominator. In this case, ( x^3 \div x = x^2 ).

Multiply the entire denominator by the result obtained in step 1. This gives ( x^2(x  1) = x^3  x^2 ).

Subtract the result obtained in step 2 from the numerator. So, ( (x^3 + x^2  x  1)  (x^3  x^2) = 2x^2  x  1 ).

Repeat steps 13 with the new polynomial obtained in step 3.

Divide the first term of the new polynomial by the first term of the denominator. Here, ( 2x^2 \div x = 2x ).

Multiply the entire denominator by the result obtained in step 5. This gives ( 2x(x  1) = 2x^2  2x ).

Subtract the result obtained in step 6 from the new polynomial. So, ( (2x^2  x  1)  (2x^2  2x) = x  1 ).

Repeat steps 13 with the new polynomial obtained in step 7.

Divide the first term of the new polynomial by the first term of the denominator. Here, ( x \div x = 1 ).

Multiply the entire denominator by the result obtained in step 9. This gives ( (x  1) = x  1 ).

Subtract the result obtained in step 10 from the new polynomial. So, ( (x  1)  (x  1) = 0 ).
Therefore, ( \frac{x^3 + x^2  x  1}{x  1} = x^2 + 2x + 1 ), with a remainder of 0.
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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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