Pre-Calculus Polynomial and rational functions

Higher-degree polynomial functions and their graphs

Q: What is multiplicity?

The exponent on a factor. x minus r to the k gives a zero at r of multiplicity k; even multiplicity makes the graph touch the axis, odd makes it cross.

Q: How many turning points can a polynomial have?

At most n minus 1, where n is the degree. It may have fewer.

Q: How do you find the end behavior?

Use the leading term only. Even degree, both ends the same way; odd degree, opposite ways. A negative leading coefficient flips the directions.

20 practice questions 0 video lessons Theory + worked examples

Higher-Degree Polynomial Functions

Texas Precalculus (TEKS) • Standard P.2(F), P.2(N) • Polynomial & Rational Functions

Higher-Degree Polynomial Functions is the opening topic of Polynomial & Rational Functions in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.2(F), P.2(N), which requires students to graph polynomial functions and analyze the situations they model.

A polynomial function is a smooth, continuous curve whose degree and leading coefficient set its end behavior, with zeros whose multiplicity decides whether the graph crosses or touches the \(x\)-axis.

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Practice questions

Every question with a fully worked solution.

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Theory

A polynomial function has the form

\[f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\qquad a_n\neq 0,\]

where \(n\) is the degree and \(a_n\) the leading coefficient. Its graph is a single smooth, continuous curve — no breaks, holes, or sharp corners.

End behavior is fixed by the leading term (degree even/odd, coefficient sign).
Zeros (roots) are where \(f(x)=0\); a factor \((x-r)^k\) gives a zero of multiplicity \(k\).
At a zero of even multiplicity the graph touches the axis; at odd multiplicity it crosses.
A degree-\(n\) polynomial has at most \(n-1\) turning points.

Reading a graph: degree bounds the turning points and zeros, the leading term sets the ends, and multiplicity tells you touch versus cross at each root.

A quartic with four zeros (gold) and three turning points.

\((x+1)^2(x-2)\): touches at the double root \(x=-1\), crosses at \(x=2\).

Standard form and the key counts:

\[f(x)=a_nx^n+\cdots+a_0,\quad \#\text{turning points}\le n-1,\quad \#\text{real zeros}\le n\]

polynomial in standard form; at most n minus 1 turning points and n zeros

A factor \((x-r)^k\) contributes a zero \(r\) of multiplicity \(k\): even \(\Rightarrow\) touch, odd \(\Rightarrow\) cross.

How to sketch a polynomial

Degree & leading term \(\Rightarrow\) end behavior.
Factor to find the zeros and their multiplicities.
Plot the zeros, marking touch (even) or cross (odd) at each.
Find the \(y\)-intercept \(f(0)\), then connect with a smooth curve consistent with the ends and turning-point limit.

Example 1 — Degree, leading term, end behavior

For \(f(x)=-2x^3+4x^2-x+5\), state the degree, leading coefficient, and end behavior.

Solution

The highest power gives the degree and leading term.

\(\text{degree}\)	\(=\)	\(3\ \text{(odd)}\)
\(\text{leading coeff.}\)	\(=\)	\(-2\ \text{(negative)}\)

Odd degree with a negative leading coefficient: up on the left, down on the right.

\(x\to-\infty\)	\(\Rightarrow\)	\(f\to+\infty\)
\(x\to+\infty\)	\(\Rightarrow\)	\(f\to-\infty\)

degree 3, leading coefficient negative 2, up-left down-right

Example 2 — Zeros and their multiplicity

Find the zeros of \(f(x)=(x+1)^2(x-2)\) and describe the graph at each.

Solution

Set each factor to zero; the exponent is the multiplicity.

\(x+1=0\)	\(\Rightarrow\)	\(x=-1\ \text{(multiplicity 2)}\)
\(x-2=0\)	\(\Rightarrow\)	\(x=2\ \text{(multiplicity 1)}\)

Even multiplicity \(\Rightarrow\) the curve touches at \(x=-1\); odd multiplicity \(\Rightarrow\) it crosses at \(x=2\).

double root at negative 1 touches; single root at 2 crosses

Example 3 — Build a polynomial from its zeros

Write a polynomial of least degree with zeros \(x=0,\ 3,\ -2\).

Solution

Each zero \(r\) contributes a factor \((x-r)\).

\(f(x)\)	\(=\)	\(x(x-3)(x+2)\)
	\(=\)	\(x(x^2-x-6)\)
	\(=\)	\(x^3-x^2-6x\)

polynomial is x cubed minus x squared minus 6x

Example 4 — Turning points

What is the greatest possible number of turning points of a degree-5 polynomial?

Solution

A degree-\(n\) polynomial has at most \(n-1\) turning points.

\(n-1\)

\(=\)

\(5-1=4\)

So at most 4 turning points.

at most 4 turning points for degree 5

Common pitfalls

Multiplicity changes the graph's shape. A double root turns the curve back; it does not pass through.

Turning points are bounded by \(n-1\), not equal to it. A degree-4 polynomial can have 1 or 3 turning points, not necessarily 3.

End behavior is only the leading term. Lower terms shift the middle but never the far-left/far-right direction.

Frequently asked questions

What is the degree of a polynomial?

The highest exponent on the variable. It bounds the number of zeros (\(\le n\)) and turning points (\(\le n-1\)) and, with the leading coefficient, fixes the end behavior.

What is multiplicity?

The exponent on a factor. \((x-r)^k\) gives a zero at \(r\) of multiplicity \(k\); even multiplicity makes the graph touch the axis, odd makes it cross.

How many turning points can a polynomial have?

At most \(n-1\), where \(n\) is the degree. It may have fewer.

How do you find the end behavior?

Use the leading term only. Even degree: both ends the same way; odd degree: opposite ways. A negative leading coefficient flips the directions.

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