Sketching polynomials from their zeroes

Reviewing the factor theorem, sketching a polynomial from its linear factors, the behaviour of the graph at zeroes of odd and even multiplicity, and the leading term at the far left and right.

This builds on the polynomials chapter from Year 11 Extension 1.

The factor theorem

$P(\alpha) = 0$ exactly when $(x - \alpha)$ is a factor of $P(x)$. A zero $\alpha$ has multiplicity $m$ when $(x - \alpha)^m$ is a factor but $(x - \alpha)^{m+1}$ is not.

Shape at a zero

Multiplicity	Behaviour of the graph
$1$ (single)	Crosses the axis, like a line.
$2$ (double)	Touches the axis and turns back, like a parabola.
$3$ (triple)	Crosses the axis while flattening out: a horizontal point of inflection.

In general, an odd multiplicity crosses and an even multiplicity touches. Far to the left and right, the graph follows its leading term.

Multiplicity changes the shape

y = 0.3(x + 1)ⁿ(x − 2). At x = −1, the graph crosses when n is odd and touches when n is even.

Try this

Which value of $n$ gives a horizontal point of inflection at $x = -1$?

Example 1●●

Sketch $y = (x + 1)^2(x - 2)$.

Zeroes: a double zero at $x = -1$ (touches) and a single zero at $x = 2$ (crosses).

The $y$-intercept is $(1)^2(-2) = -2$.

The leading term is $x^3$, so the graph rises to the right and falls to the left. It comes up to touch the axis at $x = -1$, dips through $(0, -2)$, then crosses at $x = 2$.

Example 2●●

Find a cubic with a double zero at $x = 3$, a single zero at $x = -1$, and $y$-intercept $18$.

$P(x) = a(x - 3)^2(x + 1)$.

$P(0) = a \times 9 \times 1 = 18$, so $a = 2$ and $P(x) = 2(x - 3)^2(x + 1)$.

Flashcards

Single zero

The graph crosses the axis.

Double zero

The graph touches the axis and turns back.

Triple zero

The graph crosses with a horizontal point of inflection.

Practice

Find the $y$-intercept of $y = (x + 1)^2(x - 2)$.
A cubic is $a(x - 3)^2(x + 1)$ with $y$-intercept $18$. Find $a$.
What is the multiplicity of the zero $x = 4$ of $(x - 4)^3(x + 1)$?
What is the degree of $(x - 1)^2(x + 2)^3$?
At a double zero, the graph:
At a triple zero, the graph:
As $x \to -\infty$, $y = -(x - 1)^2(x + 3)$:
Which has the graph touching the $x$-axis at $x = 2$?