Let f(x)=(2x^3-x^2+2x+1)/(x^2+1). What is the equation of the oblique asymptote of f(x)?
Can someone show me how to do the long division one for this question. I have always used synthetic division but realized there are some questions that require to do long division.
as x gets very large, the higher powers dominate, making the lower powers insignificant. So, for large x,
f(x) looks very much like 2x^3/x^2 = 2x
As x gets bigger and bigger, y gets closer and closer to just 2x.
To find the equation of the oblique asymptote of the function f(x), you can perform long division as follows:
Step 1: Divide the numerator (2x^3 - x^2 + 2x + 1) by the denominator (x^2 + 1).
_____________________________
x^2 + 1 | 2x^3 - x^2 + 2x + 1
Step 2: Divide the first term of the numerator (2x^3) by the first term of the denominator (x^2) to get 2x.
_____________________________
x^2 + 1 | 2x^3 - x^2 + 2x + 1
- (2x^3 + 2x)
Step 3: Multiply the quotient (2x) by the denominator (x^2 + 1) and subtract it from the numerator.
_____________________________
x^2 + 1 | 2x^3 - x^2 + 2x + 1
- (2x^3 + 2x)
_________________________
-3x^2 + 2x + 1
Step 4: Repeat steps 2 and 3 with the new numerator (-3x^2 + 2x + 1) until the degree of the numerator becomes lower than the degree of the denominator.
_____________________________
x^2 + 1 | 2x^3 - x^2 + 2x + 1
- (2x^3 + 2x)
_________________________
-3x^2 + 2x + 1
- (-3x^2 - 3)
________________________
5x + 4
Step 5: The resulting quotient is the equation of the oblique asymptote.
In this case, the quotient is 2x - 3. Therefore, the equation of the oblique asymptote of f(x) is y = 2x - 3.
Note: Long division is necessary when the degrees of the numerator and denominator are the same or when the degree of the numerator is higher than the degree of the denominator. Synthetic division is used when the denominator is a linear factor (degree 1).
To find the equation of the oblique asymptote of the function f(x), you can use polynomial long division. Here's a step-by-step guide on how to perform long division for this specific problem:
Step 1: Set up the division. Write the dividend (2x^3 - x^2 + 2x + 1) as the numerator and the divisor (x^2 + 1) as the denominator.
______________________________________
x^2 + 1 | 2x^3 - x^2 + 2x + 1
Step 2: Divide the first term of the dividend (2x^3) by the first term of the divisor (x^2). The result is 2x.
______________________________________
x^2 + 1 | 2x^3 - x^2 + 2x + 1
-(2x^3 + 2x)
Step 3: Multiply the divisor by the result obtained in the previous step (2x) and write the product below the dividend, aligning the like terms.
______________________________________
x^2 + 1 | 2x^3 - x^2 + 2x + 1
-(2x^3 + 2x)
_______________
-3x^2 + 2x
Step 4: Subtract the product from the dividend.
______________________________________
x^2 + 1 | 2x^3 - x^2 + 2x + 1
-(2x^3 + 2x)
_______________
-3x^2 + 2x
-(-3x^2 - 3)
Step 5: Bring down the next term of the dividend (2x + 1).
______________________________________
x^2 + 1 | 2x^3 - x^2 + 2x + 1
-(2x^3 + 2x)
_______________
-3x^2 + 2x
-(-3x^2 - 3)
_______________
5x + 4
Step 6: Repeat steps 2-5 using the current result (5x) as the new first term of the dividend.
______________________________________
x^2 + 1 | 2x^3 - x^2 + 2x + 1
-(2x^3 + 3x)
_______________
-3x^2 + 2x
-(-3x^2 - 3)
_______________
5x + 4
-(5x + 5)
Step 7: Divide the new first term of the dividend (5x) by the first term of the divisor (x^2). The result is 5.
______________________________________
x^2 + 1 | 2x^3 - x^2 + 2x + 1
-(2x^3 + 3x)
_______________
-3x^2 + 2x
-(-3x^2 - 3)
_______________
5x + 4
-(5x + 5)
________________
-1
Step 8: Multiply the divisor by the result obtained in the previous step (5) and write the product below the dividend, aligning the like terms.
______________________________________
x^2 + 1 | 2x^3 - x^2 + 2x + 1
-(2x^3 + 3x)
_______________
-3x^2 + 2x
-(-3x^2 - 3)
_______________
5x + 4
-(5x + 5)
________________
-1
-5
Step 9: Subtract the product from the dividend.
______________________________________
x^2 + 1 | 2x^3 - x^2 + 2x + 1
-(2x^3 + 3x)
_______________
-3x^2 + 2x
-(-3x^2 - 3)
_______________
5x + 4
-(5x + 5)
________________
-1
-5
_________________
-6
The remainder obtained is -6.
Step 10: Write the division result as the quotient (2x - 3) with the remainder (-6) over the divisor (x^2 + 1).
Therefore, the long division of f(x) results in:
f(x) = 2x - 3 - 6/(x^2 + 1)
The equation of the oblique asymptote is the quotient part of the division, which is 2x - 3.