Polynomial functions: zeros, multiplicity and end behaviour

What is it about?

A polynomial function that splits into linear factors can be written like this:

Here $x_1, x_2, x_3$ are the zeros, $m_1, m_2, m_3$ their multiplicities and $a \neq 0$ the leading coefficient. The degree is $n = m_1 + m_2 + m_3$. If two zeros coincide, their multiplicities add up: $(x - 1) \cdot (x - 1)^2 = (x - 1)^3$. Multiplying out gives the expanded form, for example

The y-intercept is the constant term: $f(0) = 0.5 \cdot 2 \cdot (-0.5) \cdot (-3) = 1.5$.

The last column gives the reason: near a zero $x_0$ the other factors hardly change. There, $f(x) \approx c \cdot (x - x_0)^m$, where $c$ is the product of $a$ and the other factors at $x_0$. For $f(x) = 0.2 \cdot (x + 3) \cdot x^2 \cdot (x - 3)$ near $0$ this gives $c = 0.2 \cdot 3 \cdot (-3) = -1.8$ and $f(x) \approx -1.8x^2$: a parabola opening downwards.

In the sign chart every factor gets its own row: $(x - x_0)^m$ is positive to the right of $x_0$ and $(-1)^m$ to the left, so negative for odd $m$. The sign of $f$ is the product of the column (together with the sign of $a$).

End behaviour: Factoring out $x^n$ gives $\frac{f(x)}{a \cdot x^n} = \left(1 - \frac{x_1}{x}\right)^{m_1} \cdots \to 1$ for $|x| \to \infty$. For large $|x|$ only the term $a x^n$ counts. In the example above, at $x = 100$ we already have $\frac{f(100)}{0.5 \cdot 100^3} = 1.02 \cdot 0.995 \cdot 0.97 \approx 0.984$. As $x \to +\infty$, $f(x)$ has the sign of $a$; for even $n$ both ends behave the same way, for odd $n$ they go in opposite directions.

Try it

1. Three simple zeros: Drag the beads on the $x$-axis. Tap a bead – every ring stands for one more multiplicity. How do the graph, the term and the sign chart change?
2. Compare a double, a triple and a fourfold zero. When does the sign change?
3. Push two beads onto each other until they merge. What happens to the rings and to the term? All three coincide: Which function do you get?
4. Drag the handle for $a$ on the graph up and down, also below the $x$-axis. Which end-behaviour arrows turn around? Negative a
5. Switch on the magnifier: Point at the beads one after another with the mouse (on a tablet: move a bead a little). What does the graph look like in the magnifier each time? Compare with the dashed approximations.
6. Press “View from far away”. The view zooms out by a factor of 20. Why can you hardly tell $f$ and the dashed curve $a x^n$ apart?
7. Degree 9 and degree 12: Use the sign chart to find where $f$ is negative.

Tasks

Task 1: Read off the properties

Let $f(x) = -0.4 \cdot (x + 3)(x + 0.5)(x - 2.5)$. Give the degree, the zeros, the end behaviour and $f(0)$. Load task

Task 2: Which term fits?

The graph of a cubic function touches the $x$-axis at $x = -2$, crosses it at $x = 2$ and passes through the point $(1, -4.5)$. Find the term in factored form and in expanded form. Load task

Task 3: Multiplying out

Multiply out $f(x) = 0.3 \cdot (x - 1)^3 (x + 3)$. Check with “Expand step by step”.

Task 4: Sign chart

Make a sign chart for $f(x) = 0.2 \cdot (x + 3) \cdot x^2 \cdot (x - 3)$ with one row per factor. Where is $f(x) < 0$? Calculate $f(1)$ and give the end behaviour. Load task

Task 5: Magnifier at the double zero

Explain for the function from task 4 why the graph near $x = 0$ behaves like the parabola $y = -1.8x^2$. Compare $f(0.1)$ with $-1.8 \cdot 0.1^2$.

Task 6: True or false?

a) “A polynomial function of degree 4 has four zeros.” b) “If $a > 0$, then $f(x) \to +\infty$ as $x \to -\infty$.”

Ideas for the classroom

• Use: About grade 10, polynomial functions (zeros with multiplicity, end behaviour); in upper secondary as revision before curve sketching. Please check against the curriculum and your textbook whether sign charts with one row per factor are common there.
• Guess the term: With “Show the term and properties” switched off, only the graph appears. Pairs can set up graphs for each other and find the term; “Reveal the term” resolves it.
• From the magnifier to the far view: The magnifier shows why the graph crosses or touches at a zero (it behaves like a power $c \cdot (x - x_0)^m$ there), the view from far away shows why only $a x^n$ decides the end behaviour. Both work well discussed one after the other.
• Model: Zeros at half units between $-4$ and $4$, multiplicities up to 4. Only functions that split completely into linear factors are shown; functions such as $x^2 + 1$ do not occur.
• Common stumbling blocks:
  • Sign inside the bracket: the zero of $(x + 2)$ is $-2$.
  • “The graph crosses the axis at every zero.” – For even multiplicity it only touches it.
  • For the end behaviour only $a$ is considered and the degree is forgotten (or vice versa).
  • When multiplying out, $a$ is only multiplied with the first factor.