# Graph (x^3-10x^2+32x-48)/(x-6) x3-10×2+32x-48x-6
Find where the expression x3-10×2+32x-48x-6 is undefined.
x=6
The vertical asymptotes occur at areas of infinite discontinuity.
No Vertical Asymptotes
Consider the rational function R(x)=axnbxm where n is the degree of the numerator and m is the degree of the denominator.
1. If n<m, then the x-axis, y=0, is the horizontal asymptote.
2. If n=m, then the horizontal asymptote is the line y=ab.
3. If n>m, then there is no horizontal asymptote (there is an oblique asymptote).
Find n and m.
n=3
m=1
Since n>m, there is no horizontal asymptote.
No Horizontal Asymptotes
Find the oblique asymptote using polynomial division.
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of 0.
 x – 6 x3 – 10×2 + 32x – 48
Divide the highest order term in the dividend x3 by the highest order term in divisor x.
 x2 x – 6 x3 – 10×2 + 32x – 48
Multiply the new quotient term by the divisor.
 x2 x – 6 x3 – 10×2 + 32x – 48 + x3 – 6×2
The expression needs to be subtracted from the dividend, so change all the signs in x3-6×2
 x2 x – 6 x3 – 10×2 + 32x – 48 – x3 + 6×2
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
 x2 x – 6 x3 – 10×2 + 32x – 48 – x3 + 6×2 – 4×2
Pull the next terms from the original dividend down into the current dividend.
 x2 x – 6 x3 – 10×2 + 32x – 48 – x3 + 6×2 – 4×2 + 32x
Divide the highest order term in the dividend -4×2 by the highest order term in divisor x.
 x2 – 4x x – 6 x3 – 10×2 + 32x – 48 – x3 + 6×2 – 4×2 + 32x
Multiply the new quotient term by the divisor.
 x2 – 4x x – 6 x3 – 10×2 + 32x – 48 – x3 + 6×2 – 4×2 + 32x – 4×2 + 24x
The expression needs to be subtracted from the dividend, so change all the signs in -4×2+24x
 x2 – 4x x – 6 x3 – 10×2 + 32x – 48 – x3 + 6×2 – 4×2 + 32x + 4×2 – 24x
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
 x2 – 4x x – 6 x3 – 10×2 + 32x – 48 – x3 + 6×2 – 4×2 + 32x + 4×2 – 24x + 8x
Pull the next terms from the original dividend down into the current dividend.
 x2 – 4x x – 6 x3 – 10×2 + 32x – 48 – x3 + 6×2 – 4×2 + 32x + 4×2 – 24x + 8x – 48
Divide the highest order term in the dividend 8x by the highest order term in divisor x.
 x2 – 4x + 8 x – 6 x3 – 10×2 + 32x – 48 – x3 + 6×2 – 4×2 + 32x + 4×2 – 24x + 8x – 48
Multiply the new quotient term by the divisor.
 x2 – 4x + 8 x – 6 x3 – 10×2 + 32x – 48 – x3 + 6×2 – 4×2 + 32x + 4×2 – 24x + 8x – 48 + 8x – 48
The expression needs to be subtracted from the dividend, so change all the signs in 8x-48
 x2 – 4x + 8 x – 6 x3 – 10×2 + 32x – 48 – x3 + 6×2 – 4×2 + 32x + 4×2 – 24x + 8x – 48 – 8x + 48
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
 x2 – 4x + 8 x – 6 x3 – 10×2 + 32x – 48 – x3 + 6×2 – 4×2 + 32x + 4×2 – 24x + 8x – 48 – 8x + 48 0
Since the remander is 0, the final answer is the quotient.
x2-4x+8
The oblique asymptote is the polynomial portion of the long division result.
y=x2-4x
y=x2-4x
This is the set of all asymptotes.
No Vertical Asymptotes
No Horizontal Asymptotes
Oblique Asymptotes: y=x2-4x
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