Graph (x^3-10x^2+32x-48)/(x-6)

Math
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.
Tap for more steps…
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of 0.
x6x310×2+32x48
Divide the highest order term in the dividend x3 by the highest order term in divisor x.
x2
x6x310×2+32x48
Multiply the new quotient term by the divisor.
x2
x6x310×2+32x48
+x36×2
The expression needs to be subtracted from the dividend, so change all the signs in x3-6×2
x2
x6x310×2+32x48
x3+6×2
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
x2
x6x310×2+32x48
x3+6×2
4×2
Pull the next terms from the original dividend down into the current dividend.
x2
x6x310×2+32x48
x3+6×2
4×2+32x
Divide the highest order term in the dividend -4×2 by the highest order term in divisor x.
x24x
x6x310×2+32x48
x3+6×2
4×2+32x
Multiply the new quotient term by the divisor.
x24x
x6x310×2+32x48
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
x24x
x6x310×2+32x48
x3+6×2
4×2+32x
+4×224x
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
x24x
x6x310×2+32x48
x3+6×2
4×2+32x
+4×224x
+8x
Pull the next terms from the original dividend down into the current dividend.
x24x
x6x310×2+32x48
x3+6×2
4×2+32x
+4×224x
+8x48
Divide the highest order term in the dividend 8x by the highest order term in divisor x.
x24x+8
x6x310×2+32x48
x3+6×2
4×2+32x
+4×224x
+8x48
Multiply the new quotient term by the divisor.
x24x+8
x6x310×2+32x48
x3+6×2
4×2+32x
+4×224x
+8x48
+8x48
The expression needs to be subtracted from the dividend, so change all the signs in 8x-48
x24x+8
x6x310×2+32x48
x3+6×2
4×2+32x
+4×224x
+8x48
8x+48
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
x24x+8
x6x310×2+32x48
x3+6×2
4×2+32x
+4×224x
+8x48
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
Graph (x^3-10x^2+32x-48)/(x-6)

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