No idea how to do this

3)  f(x)  =  -2x²(x - 2)³(x + 4)⁴(x - 5)

 

The degree of a polynomial is the number of times that the variable (the 'x') is used as a factor.

  in x²         =  2 times

  in (x - 2)³  =  3 times

  in (x + 4)⁴  =  4 times

  in (x - 5)    =  1 time

  Total:  2 + 3 + 4 + 1  =  10 times   <--   degree = 10

 

End behavior:  What happens at the far left-end of the graph and at the far right-end of the graph.

  For left-end behavior, place a "large" negative number (such as -10000) into each factor and see if the factor

    becomes positive or negative:

    -2           is negative

    x²           is positive

    (x - 2)³  is negative

    (x + 4)⁴  is positive

    (x - 5)    is negative

    Multiplying together, the answer is negative; so at the far left-end of the graph, the y-value will be negative;

    so the graph starts at the lower-left end.

 

  For right-end behavior, place a large positive number (such as 10000) into each factor and see if the factor

    is positive or negative:

    -2                is negative

    x²               is positive

    (x - 2)³       is positive

    (x + 4)⁴      is positive

    (x - 5)        is positive

    Multiplying together, the answer is negative; so at the far right-end of the graph, the y-value will be negative;

    so the graph ends at the lower-right end.

 

Zeros occur at the x-values that make the factors zero.

  x²                 zero at 0

  (x - 2)³         zero at 2

  (x + 4)⁴        zero at -4

  (x- 5)           zero at 5

 

Bounces or pass through:  the graph bounces at a zero that has an even degree, passes through at a zero that

  has an odd degree.

  x²                 bounces at 0

  (x - 2)³         passes through at 2

  (x + 4)⁴        bounces at -4

  (x - 5)           passes through at 5

 

4)  f(x)  =  3x⁷ - 48x⁵

 

First, factor this:  f(x)  =  3x⁷ - 48x⁵  =  3x⁵(x² - 16)  =  3x⁵(x + 4)(x - 4)

 

Now, analyze this as problem 4) was analyzed.