Knowledge Notes | Carey AP Precalculus Practice Lab

Crossing and touching sketches illustrate multiplicity only; they are not scaled plots of the example polynomial.

What a strong response shows

At a real zero, compare odd and even multiplicity.
Use conjugate pairs only for real-coefficient polynomials.
For even or odd symmetry, compare values at opposite inputs.

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What You Need to Know

Key idea

Zeros are inputs that make a polynomial output zero, and each zero corresponds to a factor.

Fast decision rule

Set the polynomial equal to zero, factor when possible, then count repeated factors.

Worked example

A real zero of multiplicity two makes the polynomial graph touch the horizontal axis and turn; the local sign does not change.

Common mistake

Reading every x-intercept as a simple zero.

Mini check

What graph clue can suggest an even multiplicity?

Learning Objectives

Key idea

1.5.A identifies polynomial characteristics from zeros; 1.5.B determines whether a polynomial is even or odd.

Fast decision rule

Use factors for zeros and symmetry tests for even or odd classification.

Worked steps

$1$ . Factor or inspect the given form. $2$ . Record each zero and multiplicity. $3$ . Use conjugate pairing for nonreal zeros. $4$ . Test $p\left(-x\right)$ for symmetry.

Common mistake

Calling a polynomial odd because it has an odd degree.

Mini check

What must be true for a polynomial to be even?

Key Concepts

Key idea

Multiplicity controls crossing or touching, and nonreal complex zeros occur in conjugate pairs when coefficients are real.

Fast decision rule

At a real polynomial zero, odd multiplicity crosses and even multiplicity touches and turns. For real coefficients, pair a nonreal zero $a+b i$ with $a-b i$ .

Worked example

If a real-coefficient polynomial has $2+3 i$ as a zero, $2-3 i$ must also be a zero.

Common mistake

Counting a conjugate pair as one zero in a degree total.

Mini check

How many degree units does a conjugate pair use?

Relevant Representations

Key idea

Factored form, a graph, and a zero list each reveal a different part of the same polynomial.

Fast decision rule

Use factored form for multiplicity, the graph for crossing behavior, and the zero list for degree counting.

Use the representation that shows the evidence
Representation	Best evidence	Check
Factored form	zero and multiplicity	set each factor to zero
Graph	crossing or touching	look near the intercept
Equation test	even or odd symmetry	replace $x$ by $-x$
Zero list	degree accounting	include conjugates

Common mistake

Using the graph alone to identify a complex zero.

Mini check

Which representation can show a nonreal zero exactly?

Essential Relationships or Formulas

Key idea

Factor, symmetry, and conjugate rules organize zero information.

p(r)=0\Longleftrightarrow (x-r)\text{ is a factor of }p(x)

a+bi\text{ is a zero}\Longrightarrow a-bi\text{ is a zero}

p(-x)=p(x)\text{ even},\qquad p(-x)=-p(x)\text{ odd}

Fast decision rule

Keep multiplicities in the factorization and count every zero toward the degree.

Worked example

A polynomial with factors $\left(x-2\right)^{2}$ and $\left(x+1\right)$ has zeros $2$ with multiplicity $2$ and $-1$ with multiplicity $1$ .

Common mistake

Dropping the squared factor after finding its zero.

Mini check

What happens to the degree count if a zero has multiplicity $3$ ?

Worked Examples

Key idea

Start from the factor structure, then translate it into graph and symmetry statements.

Fast decision rule

Find zeros first; interpret multiplicity and symmetry second.

Worked steps

$1$ . Set each factor equal to zero. $2$ . Record the exponent as multiplicity. $3$ . Predict crossing or touching. $4$ . Test $p\left(-x\right)$ if symmetry is asked. $5$ . Count degree.

Worked example

If $p\left(3\right)=-4$ is a relative maximum and $p$ is odd, then the point reflected through the origin is $\left(-3,\,4\right)$ , giving the matching behavior at the opposite input.

Common mistake

Reflecting across the y-axis for an odd function.

Mini check

Which point transformation belongs to an odd function?

Common Mistakes

Key idea

Common errors are multiplicity, conjugate, and symmetry errors.

Fast decision rule

Write the factor, exponent, graph behavior, and symmetry test in separate lines.

Worked example

A zero with multiplicity $2$ can touch the axis while a zero with multiplicity $1$ crosses it; both are zeros, but their graph behavior differs.

Common mistake

Assuming every repeated zero crosses the axis.

Mini check

What single sign in the exponent changes the likely graph behavior?

AP Exam Tips

Key idea

The fastest route is factor $\to$ zero $\to$ multiplicity $\to$ graph behavior $\to$ degree check.

Fast decision rule

Use the form shown in the prompt; do not expand unless the question asks for coefficients or symmetry.

Worked steps

$1$ . Locate factors or intercepts. $2$ . Record multiplicity. $3$ . Pair complex zeros. $4$ . Test symmetry. $5$ . Check degree and leading behavior.

Common mistake

Stopping after finding real zeros and forgetting the degree or conjugate requirement.

Mini check

Does your answer account for every degree unit?

Practice

Use the Questionbank items below to interpret polynomial zeros, multiplicity, and symmetry.

TRY IT IN A NEW SETTING

Can you transfer the method?

For $p(x)=(x+1)^3(x-2)^2$ , state the real zeros, multiplicities and crossing behavior.

Compare your reasoning

At $x=-1$ , multiplicity $3$ gives a crossing. At $x=2$ , multiplicity $2$ gives a touch and turn. Counting multiplicity gives degree $5$ .

Use it in a real question

Work independently, then compare the mathematical evidence in your answer.

Practise this topic Check the whole unit

1.5 · Polynomial Functions and Complex Zeros

Topic 1.5 · Polynomial Functions and Complex Zeros

What a strong response shows

What You Need to Know

Learning Objectives

Key Concepts

Relevant Representations

Essential Relationships or Formulas

Worked Examples

Common Mistakes

AP Exam Tips

Can you transfer the method?