Mathematical Investigations III
Name:
Mathematical Investigations III - A View of the World
"Pass Through" and "Bounce" Points
Throughout this worksheet, all polynomials are assumed to have real number coefficients.
The exponent of a factor in a polynomial is called the multiplicity of that factor. Since
each factor of ( x  r ) corresponds to a zero (or root or x-intercept) at x  r , we define the
multiplicity of the root at x  r to be the multiplicity of the factor ( x  r ) .
Example: Let y =  x  7   x  4  x  2 . Then the factor ( x  7) has multiplicity 4, the
factor ( x  4) has multiplicity 1, and the factor ( x  2) has multiplicity 3. Similarly, the
zero at x  7 has multiplicity 4, the zero at x  4 has multiplicity 1, and the zero at
x  2 has multiplicity 3.
4
3
1. We want to examine the role of the multiplicity of each factor and its effect on the graph
of the polynomial.
a) Using graphing utility (Winplot, Mathematica, etc.), make a quick sketch of the graph of
each of the following functions. Follow the sequence, use a meaningful window, and
mark the x- and y-intercepts with their values. It is not important to have the heights
drawn to scale. Draw smooth flowing curves.
b) Determine intervals on which each function is positive or negative.
y1   x  2 x  1 x  3
y2   x  2
( x  2)
( x  1)
( x  3)
2
 x  1 x  3
( x  2)2
( x  1)
( x  3)
y1
y2
Poly 5.1
Rev. S11
Mathematical Investigations III
Name:
y3   x  2 
2
 x  12  x  3
y4   x  2
( x  2)2
2
 x  1 x  33
( x  2)2
( x  1)
( x  1)2
( x  3)
( x  3)3
y3
y4
y5   x  2  x  1 x  3
3
4
y6   x  2  x  1  x  3
5
y5
3
2
y6
Poly 5.2
Rev. S11
Mathematical Investigations III
Name:
2.
Under what circumstances did you find a "pass-through" point? A "bounce" point? Explain
(clearly) the relationship between the exponents on each of the factors in the polynomial
functions and the behavior of the graphs at x = –2, 1, and 3.
3. In terms of the sign charts you made in Question 1, explain why your relationship from
Question 2 holds.
4. Determine the degree of each polynomial y1 through y6 in question 1. Recall what you wrote
on the last handout about the degree and its effect on the graph. Are the graphs on this sheet
consistent with that information? Why or why not?
Under what circumstances do graphs cross the x-axis?
5. Give the equation of a first-degree polynomial that: (If not possible, explain why.)
(a) crosses the x-axis.
(b) does not cross the x-axis.
6. Give the equation of a second-degree polynomial that:
(a) crosses the x-axis two times.
(b) touches the x-axis at exactly one point.
(c) does not touch the x-axis at all.
Poly 5.3
Rev. S11
Mathematical Investigations III
Name:
7. Now try this same idea for third-degree polynomials. Write the equation of a third-degree
polynomial that:
(a) intersects the x-axis three times.
(b) intersects the x-axis twice.
(c) intersects the x-axis once.
(d) does not intersect the x-axis.
8. Repeat using fourth-degree polynomials. Write the equation of a fourth-degree polynomial that:
(a) intersects the x-axis four times.
(b) intersects the x-axis three times.
(c) intersects the x-axis twice.
(d) intersects the x-axis once.
(e) does not intersect the x-axis.
9. Try to generalize the observations above. If P (x ) is a polynomial with degree d, for what
values of m is it possible that P (x ) intersects the x-axis m times. It might help to consider the
cases where d is odd and d is even separately.
Poly 5.4
Rev. S11
Mathematical Investigations III
Name:
10. Look back to the graphs you found in Question 1. Look at where a graph was “flat” at an
x-intercept. How can you tell from the equation of a polynomial where its “flat spots” will
be?
Here are some informal theorems based upon your observations.
Be sure you understand and can cite examples of each of them.
For any polynomial function P(x), we know that:
• the number of zeroes (or roots or solutions) is at most the degree of P (x ) unless P (x ) = 0.
• if r is a zero (root) of a polynomial, then ( x – r ) is a factor of the polynomial.
• if ( x – r ) is a factor of a polynomial, then r is a zero (root) of the polynomial.
• if m represents the multiplicity of a linear factor of P(x), then:
If m is even, then the corresponding x-intercept is a "bounce point".
If m is odd, then the corresponding x-intercept is a "pass through" point.
If m  2 , then the corresponding x-intercept occurs with a "flat spot".
• polynomials with odd degree start below the x-axis and end above the x-axis or start above the
x-axis and end below the x-axis, i.e., they start low and end high or start high and end low.
• polynomials with even degree start and end above the x-axis or start and end below the x-axis,
i.e., they start low and end low or start high and end high.
• polynomial functions are continuous. Describe what you think is meant by "continuous."
Poly 5.5
Rev. S11
Mathematical Investigations III
Name:
A few inequalities to try:
Remember the technique of graphing the zeros, and testing regions to solve each inequality.
11. Solve: ( x – 4 ) 3 ( x + 2 )2  0
12. Solve: ( x – 7 )( x + 3 )2( x + 1 )  0
13. Solve: ( x + 2 )3( x – 1 ) 2( x – 4 )  0
Poly 5.6
Rev. S11