also the x intercept is not simply found by setting y o but

also the x intercept is not simply found by setting y = o but rather factoring first

Find the x-intercepts of y = x^6 - 5x_5 + 8x_4 - 4x_3. Describe the behavior of the graph at each intercept. What happens to y as x rightarrow plusminus infinity?

Solution

y = x^6 - 5x^5 + 8x^4 - 4x^3

we have x^3 in all the terms, so take x^3 common

y = x^3 ( x^3 - 5x^2 +8x -4)

now x^3 -5x^2 +8x - 4 can be reduced .

cleary we can (x-1) is a factor , just plug x=1

1 - 5 +8 -4 = 9-9 =0

x^3 -5x^2 +8x -4 = (x-1) (x^2 - 4x +4)

now again (x^2 -4x +4) can be reduced to (x-2)^2

so finally we can y =x^3 (x-1) (x-2) (x-2)

now to find x intercept just plug y=0

0 = x^3(x-1)(x-2)(x-2)

this can be true when x= 0 or (x-1) = 0 or (x-2) =0

x=0 or x=1 or x=2

the graph these points (0,0) (1,0) (2,0)

at these intercepts it is a straight line

the curve doen\'t cross the x axis at these intercepts

as x goes to infinty , y also goes to infinty