Polynomial Equations

Question

f(x)= −8x^4−16x^3-2x^2-4x-8/x^4+2x^3+x^2+8x-9

1) What is the equation of the horizontal asympotote? y= ?

2) Does the graph of f(x)f(x) intersect its horizontal asymptote? (yes or no)

3) At what x-values does f(x)f(x) intersect its horizontal asymptote? Give your answers in increasing order.

Answer 1

highest degreed terms dominate as x gets very large

so horizontal asymptote is ratio of the highest powered terms' coefficients = -8

when those terms are equal in degree (assuming you left off parentheses)

the curve intersects the horizontal asymptote in quadrant IV, about (1.19139,-8)

also in quadrant III, about (-11.19139, -8)

just graph the curve and graph y=-8 and read off the intersection points

Desmos gives you x coordinates with 5 decimal places

Answer 2

1) f(x) = (- 8x⁴ - 16x³ - 2x² - 4x - 8)/(x⁴ + 2x³+ x² +8x - 9);

y =lim f(x) = - 8/1 = - 8, y = - 8 is H.A.

x→±∞

2) f(x) = - 8; - 8 x⁴ - 16x³ - 2x² - 4x - 8 = - 8x⁴ -16x³ - 8x² - 64x + 72; 6x² + 60x - 80 = 0

3x² + 30x - 40 = 0; x = (- 15 ± √225 + 120)/3 = (-15 ± √345)/3. So, we have 2 intersecion points

3) These points: (- 15 + √345)/3, - 8) or (- 15 - √345/3, - 8)