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Section 9.4 More Polynomial Functions
Subsection 9.4.1 Multiplicity of Roots (Order)
Some polynomial functions have roots which are repeated.
Example 9.4.1 . Quadratic with a Double Root.
Recall that for quadratic polynomials, sometimes, after factoring, the two factors are the same. For example,
\begin{align*}
f(x) \amp = x^2 - 6x + 9\\
\amp = (x-3)(x-3)\\
\amp = (x - 3)^2
\end{align*}
Then,
\(x = 3\) is said to be a
double root . At this root, the graph doesn’t cross the
\(x\) -axis, but instead just touches it and turns around (
Graph of \(y = (x-3)^2\) ).
More generally, a polynomial function can have a double root, triple root, etc.
Example 9.4.2 . Polynomial Graphs with Repeated Roots.
\(y = (x + 1)(x - 2)(x - 3)\text{.}\) This has zeros at
\(-1\text{,}\) 2, and 3. All of the other functions will have the same zeros as well.
\(y = (x + 1)^2(x - 2)(x - 3)\text{.}\) This has a power of 2 on the
\(x+1\text{,}\) but all of the
\(x\) -intercepts are the same. What is different about the graph at
\(x = -1\text{?}\)
\(y = (x + 1)^2(x - 2)^2(x - 3)\text{.}\) Here, there is a power of 2 on
\(x =-1\) and
\(x=2\text{.}\) What is common to both these
\(x\) -intercepts?
\(y = (x + 1)^3(x - 2)(x - 3)\text{.}\) What does the graph do when the power is 3?
\(y = (x + 1)^4(x - 2)(x - 3)\text{.}\) What does the graph do when the power is 4?
Try higher exponents on the factor
\(x+1\text{.}\) You may need to zoom out or zoom in to see the behavior at
\(x = -1\text{.}\)
Observe that for each exponent (1, 2, 3, 4, ...), the graph behaves differently at the
\(x\) -intercept.
In particular, for even exponents, the function touches and turns around. For odd exponents, the function flattens out and crosses.
The exponent associated with the root affects the graph at that
\(x\) -intercept. We call this exponent the
multiplicity of the root.
Definition 9.4.3 .
A root
\(a\) of a polynomial is of
multiplicity \(k\) if the factor
\(x-a\) is to the
\(k\) th power in the factored form of the polynomial.
In some textbooks, multiplicity is instead referred to as
order .
Intuitively, “multiplicity” comes from the word “multiple”, meaning “how many times something occurs”.
For
even multiplicity, the graph
touches the
\(x\) -axis and
turns around (or “bounces”).
For
odd multiplicity, the graph
crosses the
\(x\) -axis. Further, for a zero of multiplicity greater than 1, the graph of
\(f\) will
flatten out near the zero.
Table 9.4.4. Multiplicity
1
Even
Odd
Crosses
Touches
Crosses
Near a zero of multiplicity
\(k\text{,}\) the graph looks like the power function
\(y = x^k\text{.}\)
Multiplicity 1 looks like a straight line.
Multiplicity 2 looks like a parabola.
Multiplicity 3 looks like a cubic “S-shape”.
