Solve the following problem step by step

Question

Find the coordinates of the point where the graph of f(x)= (2x³ –4x² –10x+12)/(x²-11) crosses its slant asymptote and explain the prócedure thoroughly.

Answer 1

Since there is no x term in the denominator, we can see that the slant asymptote is

y = 2x - 4

Setting 2x - 4 = (2x^3 - 4x^2 - 10x + 12)/(x^2 - 11), we get

(2x - 4)(x^2 - 11) = 2x^3 - 4x^2 - 10x + 12

2x^3 - 22x - 4x^2 + 44 = 2x^3 - 4x^2 - 10x + 12

-22x + 44 = -10x + 12

32 = 12x

x = 8/3

y = 2(8/3) - 4 = 4/3

The curve crosses the slant asymptote at (8/3, 4/3)