How do you find the asymptotes for (x^4-3x^3-21x^2+43x+60)/(x^4-6x^3+x^2+24x-20)x43x321x2+43x+60x46x3+x2+24x20?

1 Answer
Jul 29, 2016

Horizontal asymptote (left and right): y = 1y=1

Vertical asymptotes at x=1x=1, x=2x=2, x=-2x=2

Hole at (5, 9/7)(5,97)

Explanation:

Let's attempt to factorise the numerator and denominator first.

color(white)()
Numerator

Substituting x=-1x=1 in the numerator we get:

1+3-21-43+60 = 01+32143+60=0

So x=-1x=1 is a zero and (x+1)(x+1) a factor:

x^4-3x^3-21x^2+43x+60 = (x+1)(x^3-4x^2-17x+60)x43x321x2+43x+60=(x+1)(x34x217x+60)

Substituting x=3x=3 in the remaining cubic, we get:

27-36-51+60 = 0273651+60=0

So x=3x=3 is a zero and (x-3)(x3) a factor:

x^3-4x^2-17x+60=(x-3)(x^2-x-20) = (x-3)(x-5)(x+4)x34x217x+60=(x3)(x2x20)=(x3)(x5)(x+4)

In summary:

x^4-3x^3-21x^2+43x+60 = (x+1)(x-3)(x-5)(x+4)x43x321x2+43x+60=(x+1)(x3)(x5)(x+4)

color(white)()
Denominator

Substituting x=1x=1 in the denominator we get:

1-6+1+24-20 = 016+1+2420=0

So x=1x=1 is a zero and (x-1)(x1) a factor:

x^4-6x^3+x^2+24x-20 = (x-1)(x^3-5x^2-4x+20)x46x3+x2+24x20=(x1)(x35x24x+20)

The remaining cubic factors by grouping:

x^3-5x^2-4x+20x35x24x+20

=(x^3-5x^2)-(4x-20)=(x35x2)(4x20)

=x^2(x-5)-4(x-5)=x2(x5)4(x5)

=(x^2-4)(x-5)=(x24)(x5)

=(x-2)(x+2)(x-5)=(x2)(x+2)(x5)

In summary:

x^4-6x^3+x^2+24x-20 = (x-1)(x-2)(x+2)(x-5)x46x3+x2+24x20=(x1)(x2)(x+2)(x5)

color(white)()
Together

(x^4-3x^3-21x^2+43x+60)/(x^4-6x^3+x^2+24x-20)x43x321x2+43x+60x46x3+x2+24x20

=((x+1)(x-3)color(red)(cancel(color(black)((x-5))))(x+4))/((x-1)(x-2)(x+2)color(red)(cancel(color(black)((x-5)))))

=((x+1)(x-3)(x+4))/((x-1)(x-2)(x+2))

with exclusion x != 5

Looking at the leading terms of the numerator and denominator, the degrees and coefficients are identical, so there is a horizontal asymptote (left and right): y = 1

When x=1 or +-2 the denominator is zero and the numerator non-zero, so these values of x correspond to vertical asymptotes.

Substituting the value x=5 into the simplified rational function, we find the coordinate of the hole as: (5, 9/7)

graph{(x^4-3x^3-21x^2+43x+60)/(x^4-6x^3+x^2+24x-20) [-41.83, 38.17, -10.24, 29.76]}