Graph (x^3-1)/(x^2-9)

Math
x3-1×2-9
Find where the expression (x-1)(x2+x+1)(x+3)(x-3) is undefined.
x=-3,x=3
Since (x-1)(x2+x+1)(x+3)(x-3)→-∞ as x→-3 from the left and (x-1)(x2+x+1)(x+3)(x-3)→∞ as x→-3 from the right, then x=-3 is a vertical asymptote.
x=-3
Since (x-1)(x2+x+1)(x+3)(x-3)→-∞ as x→3 from the left and (x-1)(x2+x+1)(x+3)(x-3)→∞ as x→3 from the right, then x=3 is a vertical asymptote.
x=3
List all of the vertical asymptotes:
x=-3,3
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=2
Since n>m, there is no horizontal asymptote.
No Horizontal Asymptotes
Find the oblique asymptote using polynomial division.
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Simplify the expression.
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Simplify the numerator.
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Rewrite 1 as 13.
x3-13×2-9
Since both terms are perfect cubes, factor using the difference of cubes formula, a3-b3=(a-b)(a2+ab+b2) where a=x and b=1.
(x-1)(x2+x⋅1+12)x2-9
Simplify.
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Multiply x by 1.
(x-1)(x2+x+12)x2-9
One to any power is one.
(x-1)(x2+x+1)x2-9
(x-1)(x2+x+1)x2-9
(x-1)(x2+x+1)x2-9
Simplify the denominator.
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Rewrite 9 as 32.
(x-1)(x2+x+1)x2-32
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=x and b=3.
(x-1)(x2+x+1)(x+3)(x-3)
(x-1)(x2+x+1)(x+3)(x-3)
(x-1)(x2+x+1)(x+3)(x-3)
Expand (x-1)(x2+x+1).
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Apply the distributive property.
x(x2+x+1)-1(x2+x+1)(x+3)(x-3)
Apply the distributive property.
x(x2+x)+x⋅1-1(x2+x+1)(x+3)(x-3)
Apply the distributive property.
x⋅x2+x⋅x+x⋅1-1(x2+x+1)(x+3)(x-3)
Apply the distributive property.
x⋅x2+x⋅x+x⋅1-1(x2+x)-1⋅1(x+3)(x-3)
Apply the distributive property.
x⋅x2+x⋅x+x⋅1-1×2-1x-1⋅1(x+3)(x-3)
Reorder x and 1.
x⋅x2+x⋅x+1x-1×2-1x-1⋅1(x+3)(x-3)
Raise x to the power of 1.
x⋅x2+x⋅x+1x-1×2-1x-1⋅1(x+3)(x-3)
Use the power rule aman=am+n to combine exponents.
x1+2+x⋅x+1x-1×2-1x-1⋅1(x+3)(x-3)
Add 1 and 2.
x3+x⋅x+1x-1×2-1x-1⋅1(x+3)(x-3)
Raise x to the power of 1.
x3+x⋅x+1x-1×2-1x-1⋅1(x+3)(x-3)
Raise x to the power of 1.
x3+x⋅x+1x-1×2-1x-1⋅1(x+3)(x-3)
Use the power rule aman=am+n to combine exponents.
x3+x1+1+1x-1×2-1x-1⋅1(x+3)(x-3)
Add 1 and 1.
x3+x2+1x-1×2-1x-1⋅1(x+3)(x-3)
Multiply x by 1.
x3+x2+x-1×2-1x-1⋅1(x+3)(x-3)
Multiply -1 by 1.
x3+x2+x-1×2-1x-1(x+3)(x-3)
Move x.
x3+x2-1×2+x-1x-1(x+3)(x-3)
Subtract 1×2 from x2.
x3+0+x-1x-1(x+3)(x-3)
Add x3 and 0.
x3+x-1x-1(x+3)(x-3)
Subtract 1x from x.
x3+0-1(x+3)(x-3)
Add x3 and 0.
x3-1(x+3)(x-3)
x3-1(x+3)(x-3)
Expand (x+3)(x-3).
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Apply the distributive property.
x3-1x(x-3)+3(x-3)
Apply the distributive property.
x3-1x⋅x+x⋅-3+3(x-3)
Apply the distributive property.
x3-1x⋅x+x⋅-3+3x+3⋅-3
Reorder x and -3.
x3-1x⋅x-3x+3x+3⋅-3
Raise x to the power of 1.
x3-1x⋅x-3x+3x+3⋅-3
Raise x to the power of 1.
x3-1x⋅x-3x+3x+3⋅-3
Use the power rule aman=am+n to combine exponents.
x3-1×1+1-3x+3x+3⋅-3
Add 1 and 1.
x3-1×2-3x+3x+3⋅-3
Multiply 3 by -3.
x3-1×2-3x+3x-9
Add -3x and 3x.
x3-1×2+0-9
Subtract 9 from 0.
x3-1×2-9
x3-1×2-9
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of 0.
x2+0x9x3+0x2+0x1
Divide the highest order term in the dividend x3 by the highest order term in divisor x2.
x
x2+0x9x3+0x2+0x1
Multiply the new quotient term by the divisor.
x
x2+0x9x3+0x2+0x1
+x3+09x
The expression needs to be subtracted from the dividend, so change all the signs in x3+0-9x
x
x2+0x9x3+0x2+0x1
x30+9x
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
x
x2+0x9x3+0x2+0x1
x30+9x
+9x
Pull the next term from the original dividend down into the current dividend.
x
x2+0x9x3+0x2+0x1
x30+9x
+9x1
The final answer is the quotient plus the remainder over the divisor.
x+9x-1×2-9
The oblique asymptote is the polynomial portion of the long division result.
y=x
y=x
This is the set of all asymptotes.
Vertical Asymptotes: x=-3,3
No Horizontal Asymptotes
Oblique Asymptotes: y=x
Graph (x^3-1)/(x^2-9)

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