Find the zeros for the given polynomial function and give th

Find the zeros for the given polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero. F(x) = x^3 - 4x^2 + 4x The zeros The multiplicity at the leftmost zeros is Determine whether the graph crosses x-axis or molecules the x-axis and truss at the leftmost zero

Solution

x³- 4x² + 4x = 0 can be written as x[x² - 2x - 2x + 4] = 0

=> x[x( x - 2) - 2(x - 2)] = 0

thus x(x - 2) ( x - 2) = 0

this is true for x = 2 and x = 0

therefore the zeros are 0,2

hence the multiplicity of leftmost zero is 1 (it just appears once in our solution) and the multiplicity of 2 is 2 (since our solution gave \'x - 2\' twice).

To find out whether the graph overshoots the x axis, differentiate the given function. This will give:

f \' (x) = 3x² - 8x + 4

for finding a maxima, the this derivative must be zero.

so 3x² - 8x + 4 = 0

whose solutions are: x = 2 and x = 2/3

substitute these values in original function to see if f(x) < 0

f(2) = 0

f(2/3) = (2/3)³- 4(2/3)² + 4(2/3) = 1.185

thus the graph just touches the x axis at x = 2 and never goes below it.