Dr. Byrne
Fall 2010
Math 112
Worksheet 4.2: Sketching Polynomials
Sketch the following polynomials by finding the zeros of the function and the intervals over
which the polynomial is negative or positive on the real line.
Note: A sketch is adequate for this worksheet if the sketch accurately represents the location of
the zeros, the y-intercept and the general shape (intervals of positive and negative sign).
f ( x)  x 6  x 4
f ( x)  ( x  4) 2 x  1
the leading term:
f ( x) 
1 3
x  2x 2
2
f ( x)  2( x  1) 3
the leading term:
Worksheet 4.2: Relationship between zeros and bends.
Draw a continuous, smooth function that intercepts the x-axis in exactly 4 locations.
With as many ‘bends’ as you can fit:
With as few ‘bends’ as you can fit:
The number of zeros a polynomial has places an [ upper ] / [lower] bound on the number of
bends a polynomial has:
A polynomial with n zeros will have ________________ bends.
Worksheet 4.2: Intermediate Value Theorem
Suppose you know the following points are on the
graph of a continuous function. Where do you know
there must there be a zero?
There must be a zero:
Intermediate Value Theorem: For a real polynomial function P(x), suppose that P(a) and P(b)
are of opposite signs. Then the function has a real zero between a and b.
Example: Use the intermediate value theorem to prove that P( x)  x 2  2 has a zero between x=1
and x=2.