Question

QM Q2.) Determine whether the graph crosses the​ x-axis, or touches the​ x-axis and turns around at each​ x-intercept.

Answer 1

f(x)=x^3+8x^2-4x-32
note that f(2)=0 so x=2 is one of the zero
dividing f(x) by (x-2)
the quotient is x^2+10x+16 which factors into (x+2)(x+8)
combining f(x)=(x-2)(x+2)(x+8)
the zeros are -8, -2 and 2

for f(x) to "turn around", it must be a max or min
f'(x)=3x^2+16x-4=0

at greatest zero, f'(2)<>0 so it crosses the x-axis

at second greatest zero, f'(-2)<>0 so it crosses the x-axis

at smallest zero, f'(-8)<>0 so it crosses the x-axis

Answer 2

by plotting y=x^3+8x^2-4x-32, the zeros are at x=-8, -2 and 2

at all three zeros, it crosses the x-axis. theres no touch n turn around.

see attached plot