what would be the obleque asymptote to f(x)=(x^7-9x^6+19x^5+23x^4-108x^3+82x^2+64x-120)/(x+3)
To find the oblique asymptote of the given function f(x), you need to perform polynomial long division. The oblique asymptote occurs when the degree of the numerator is one greater than the degree of the denominator.
First, divide the numerator (x^7-9x^6+19x^5+23x^4-108x^3+82x^2+64x-120) by the denominator (x+3) using long division:
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x + 3 | x^7 - 9x^6 + 19x^5 + 23x^4 - 108x^3 + 82x^2 + 64x - 120
- (x^7 + 3x^6)
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- 12x^6 + 19x^5 + 23x^4 - 108x^3 + 82x^2 + 64x
- (- 12x^6 - 36x^5)
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55x^5 + 23x^4 - 108x^3 + 82x^2 + 64x
- (55x^5 + 165x^4)
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- 142x^4 - 108x^3 + 82x^2 + 64x
- (- 142x^4 - 426x^3)
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318x^3 + 82x^2 + 64x - 120
- (318x^3 + 954x^2)
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-872x^2 + 64x - 120
- (-872x^2 - 2616x)
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2680x - 120
- (2680x + 8040)
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-8160
The polynomial long division gives us the quotient: x^6 - 12x^5 + 55x^4 - 142x^3 + 318x^2 - 872x + 2680, with a remainder of -8160.
Therefore, the oblique asymptote of the function f(x) is y = x^6 - 12x^5 + 55x^4 - 142x^3 + 318x^2 - 872x + 2680.