Determine the maximum number of real zeros and the maximum number of turning points for the function f(x) = 5x^3 – 4 + x – 7x^6 .

Question

Need help determining what the max. of real zeros is and the turning point.

Answer 1

Max number of complex zeros is the highest power of x (called the DEGREE of the polynomial)

Complex roots come in pairs, so if there are roots with imaginary parts, there are either 0 or 2 or 4 or 6 of them.

So the maximum number of real roots is either 6 or 4 or 2 or 0, so the maximum, in general would be 6.

For this particular polynomial, GRAPH it and count how many times the graph crosses the x-axis.  These are the REAL zeros.

 

I graphed it on desmos, and I think you can find it here:   https://www.desmos.com/calculator/va90mhvo4z

As you can see (if you can see it), this graph never crosses the x-axis so it has NO REAL ZEROS.

 

The maximum number of turning points is the highest power of x MINUS 1, or in math words:  the DEGREE - 1.  

Any 6th degree polynomial has a maximum number of turning points of 6-1 = 5 turning points.

Looking at this graph, it looks like there is only 1 turning point.

Answer 2

f(x) = -7x^6 + 5x^3 + x - 4

 

P ={+-1, +-4 }

Q = {+-1, +-7}

 

The only possible rational zeros are P/Q = {+-1. +-4, +-1/7, +-4/7}

 

None of them work; That is F(x) is not zero for any x in P/Q

So there are no rational zeros.

 

Since there are 2 sign changes, there are either zero or two positive real zeros per Decartes Rule of Signs