How do you divide #(3x^4 + 2x^3 - 11x^2 - 2x + 5)/(x^2 - 2)#?
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
Choose the first term
Multiply the divisor by
#3x^4+2x^3-11x^2-2x+5 = (x^2-2)(3x^2+2x-5) + 2x-5#
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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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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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