Identify repeated zeros and their graphical behaviour

A repeated zero is a real solution $r$ occurring with multiplicity $m$ in a factor $(x-r)^m$ ; the multiplicity records how many times the factor occurs, not multiple distinct $x$ -intercepts. An odd multiplicity makes the polynomial cross the $x$ -axis at $r$ , whereas an even multiplicity makes it touch and turn, with larger multiplicities often producing a flatter appearance near the intercept. The focus is on factored polynomial forms and real zeros, excluding complex zeros and more formal generalizations.

Detailed Explanation: Identify repeated zeros and their graphical behaviour

A zero is an $x$ -value that makes the polynomial equal to $0$ . In factored form, the exponent on a factor tells you the zero’s multiplicity.

An odd multiplicity means the graph crosses the $x$ -axis.
An even multiplicity means the graph touches the $x$ -axis and turns around.
A larger multiplicity usually makes the graph look flatter near the zero.
A repeated zero is one zero with a factor occurring more than once, not several different intercepts.

Example

Analyze the real zeros and describe the graph of

f(x)=-2(x+1)^3(x-2)^4(x-5).

Step 1: Set each factor equal to zero.

x+1=0 \quad\Rightarrow\quad x=-1

x-2=0 \quad\Rightarrow\quad x=2

x-5=0 \quad\Rightarrow\quad x=5

So the real zeros are $x=-1$ , $x=2$ , and $x=5$ .

Step 2: Read the multiplicity from each exponent.

$(x+1)^3$ gives $x=-1$ with multiplicity $3$ .
$(x-2)^4$ gives $x=2$ with multiplicity $4$ .
$(x-5)$ gives $x=5$ with multiplicity $1$ .

The zero $x=-1$ is one intercept, even though its factor occurs three times. Similarly, $x=2$ is one intercept with multiplicity $4$ .

Step 3: Use the parity of each multiplicity.

Zero	Multiplicity	Graph behavior
$x=-1$	$3$	Crosses the $x$ -axis; the odd multiplicity can make it look somewhat flat
$x=2$	$4$	Touches the $x$ -axis and turns; the even multiplicity makes it flatter
$x=5$	$1$	Crosses the $x$ -axis normally

Therefore, the graph crosses at $x=-1$ and $x=5$ , but touches and turns at $x=2$ . The zero at $x=2$ has the larger multiplicity, so the graph is especially flat near that intercept.

Learn by doing: Identify repeated zeros and their graphical behaviour

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Function Root Behaviour (Polynomials) - Roots and Multiplicity to Graph

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Identify repeated zeros and their graphical behaviour

Detailed Explanation: Identify repeated zeros and their graphical behaviour

Learn by doing: Identify repeated zeros and their graphical behaviour

Practice with unlimited practice problems

Practice:

Function Root Behaviour (Polynomials) - Roots and Multiplicity to Graph