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

Common Core Pre-Calculus • Standard F-IF.7c • Polynomial & Rational Functions

Higher-Degree Polynomial Functions is the opening topic of Polynomial & Rational Functions in the Common Core State Standards. It is aligned to Standard F-IF.7c, which requires students to graph polynomial functions, identifying zeros, multiplicity, and end behavior.

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.

Practice 20 questions

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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