Objectives
– Use factoring to find zeros of polynomials.
– Identify zeros & their multiplicities.
– Use Intermediate Value Theorem.
– Understand relationship between degree &
turning points.
Polynomial Functions
and Their Graphs
Zeros and Multiplicity
X-Intercepts (Real Zeros)
Zeros of polynomials
• When f(x) crosses (or touches) the x-axis.
• Zero- another way of saying solution
polynomial function of degree n will
have at most n x-intercepts (real zeros).
• A
• How can you find them?
– Let f(x)=0 and solve.
– Graph f(x) and see where it crosses the x-axis.
Zeros that Touch the x-axis
What if f(x) just touches the x-axis, doesn’t
cross it, then turns back up (or down) again?
This indicates f(x) did not change from
positive or negative (or vice versa), the zero
therefore exists from a square term (or some
even power). We say this has a multiplicity of
2 (if squared) or 4 (if raised to the 4th power).
•
Note: Not all zeros of a polynomial
function are real – some of the zeros are
imaginary.
Zeros and Multiplicity
f(x) = (x + 2)²(x − 1)³
Determine the zeros and
multiplicity of each zero.
1
Zeros and Multiplicity
Multiplicity and x-Intercepts
• If r is a zero of even multiplicity, then the
graph touches the x-axis and turns around at
r.
• If r is a zero of odd multiplicity, then the
graph crosses the x-axis at r.
• Regardless of whether the multiplicity of a
zero is even or odd, graphs tend to flatten out
at zeros with multiplicity greater than one.
f(x)= x³(x + 2)4(x − 3)5
Determine the zeros and
multiplicity of each zero.
Multiplicity
Example
k
If (x(x-c) is a factor of a polynomial function P(x) where
k > 1, and:
K is even
K is odd
The graph is tangent
to the xx-axis at (c, 0)
The graph crosses
the xx-axis at (c, 0)
• Find all zeros of f(x) and state the multiplicity of each zero. State
whether the graph crosses the x-axis, or touches the x-axis and
turns around, at each zero.
• f(x) = 3(x + 5)(x + 2)²
Example
• Find all zeros of f(x) and state the multiplicity of each zero. State
whether the graph crosses the x-axis, or touches the x-axis and
turns around, at each zero.
• f(x) = x³ + 5x² – 9 x – 45
2
Example
Intermediate Value Theorem
• If f(x) is positive (above the x-axis) at some point
and f(x) is negative (below the x-axis) at another
point, then f (x) = 0 (on the x-axis) at some point
between those 2 pts.
• True for any polynomial.
Turning points of a polynomial
• If a polynomial is of degree “n”, then it has at
most n-1 turning points.
• Graph changes direction at a turning point.
• These turning points are relative maxima /
relative minima.
• Use the Intermediate Value Theorem to show that the polynomial
function below has a real zero between the given integers.
• f(x)= x³ - 4x² + 2; between 0 and 1
The graph of f(x)=x5- 6x3+8x+1 is shown below. The
graph has four smooth turning points. The polynomial
is of degree 5. Notice that the graph has four turning
points.
In general, if the function is a polynomial function of
degree n, then the graph has at most n-1 turning points.
Example
Determine number of relative maxima/minima.
4
f(x) = x
3
+ 3x
2
– 2x
+ 1
3