How do you divide #(3x^4 + 2x^3 - 11x^2 - 2x + 5)/(x^2 - 2)#?

Answer 1

Use long division of the coefficients to find:

#3x^4+2x^3-11x^2-2x+5 = (x^2-2)(3x^2+2x-5) + 2x-5#

That is:

#(3x^4+2x^3-11x^2-2x+5)/(x^2-2) = 3x^2+2x-5 + (2x-5)/(x^2-2)#

Long divide the coefficients, using a long division similar to division of integers:

Note that the divisor is #1, 0, -2# to represent #x^2+0x-2# including the term in #x#.

Choose the first term #3# of the quotient to match the leading term of the dividend when multiplied by the divisor.

Multiply the divisor by #3# to get #3, 0, -6# and subtract from the dividend to get a remainder. Bring down the next term from the dividend alongside it and then repeat to get the next term of the quotient, etc. Stop when the degree of the running remainder is less than the divisor.

#3x^4+2x^3-11x^2-2x+5 = (x^2-2)(3x^2+2x-5) + 2x-5#

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Answer 2

To divide (3x^4 + 2x^3 - 11x^2 - 2x + 5) by (x^2 - 2), you can use polynomial long division.

First, divide the highest degree term of the numerator (3x^4) by the highest degree term of the denominator (x^2). This gives you 3x^2.

Next, multiply the entire denominator (x^2 - 2) by the result (3x^2), which gives you 3x^4 - 6x^2.

Subtract this result from the original numerator (3x^4 + 2x^3 - 11x^2 - 2x + 5) to get 2x^3 + 5x^2 - 2x + 5.

Repeat the process by dividing the highest degree term of this new numerator (2x^3) by the highest degree term of the denominator (x^2). This gives you 2x.

Multiply the entire denominator (x^2 - 2) by the result (2x), which gives you 2x^3 - 4x.

Subtract this result from the new numerator (2x^3 + 5x^2 - 2x + 5) to get 5x^2 + 2x + 5.

Now, divide the highest degree term of this new numerator (5x^2) by the highest degree term of the denominator (x^2). This gives you 5.

Multiply the entire denominator (x^2 - 2) by the result (5), which gives you 5x^2 - 10.

Subtract this result from the new numerator (5x^2 + 2x + 5) to get 2x + 15.

Since the degree of the new numerator (2x + 15) is less than the degree of the denominator (x^2 - 2), you have reached the end of the division.

Therefore, the quotient is 3x^2 + 2x + 5, and the remainder is 2x + 15.

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Answer from HIX Tutor

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.

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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