Question

Lef f(x) = -(x+4)(x-1)^3(x-3)^2 = -x^6 + 5x^5 + 6x^4 - 74x^3 + 151x^2 - 123x + 36

1.)Determine the end behavior

2.) Find the real zeros of f(x) & state the multiplicity of each zero. Determine the manner in which the graph of f(x) crosses or touches the x-axis at each zero.

3.) Find the x-intercepts and the y-intercept

4.) Draw the graph of f(x)

Please show work for answers- thanks :)

Answer 1

f(x) = - ( x+4)(x-1)^3 (x-3)^2

1) since the degree of polynomial is even and leading coefficient is negative

the graph opens downwards

end behaviour is

as x---> - infinity , f(x) ---> - infinity

as x ----> + infinity , f(x) ----> - infinity

2) setting each factor to 0

- ( x+4)(x-1)^3 (x-3)^2 = 0

x + 4 = 0

x = - 4

x - 1 = 0

x = 1

x - 3 = 0

x = 3

zeros are x = - 4 ( multiplicity 1 ) , x = 1 ( multiplicity 3 ) , x = 3 ( multiplicity 2 )

at x = - 4 graph crosses the x axis , at x = 1 graph crosses the x axis , at x = 3 graph touches the x axis

3) x intercepts ( -4,0) , ( 1,0) , ( 3,0)

y intercept is   -(0+4)(0-1)^3(0-3)^2

= ( 0 , 36)  

4) graph shown below