The Central Idea
Let \(P(x)\) be a polynomial and let \(c\) be a number.
The following statements are equivalent:
\[ \boxed{P(c)=0} \] \[ \boxed{c\text{ is a zero of }P} \] \[ \boxed{c\text{ is a root of }P(x)=0} \] \[ \boxed{x-c\text{ is a factor of }P(x)}. \]If \(c\) is real, then these statements are also equivalent to saying that the graph of \(y=P(x)\) has an \(x\)-intercept at
\[ \boxed{(c,0)}. \]Zero of a Polynomial Function
A number \(c\) is a zero of a polynomial function \(P\) when
\[ P(c)=0. \]For example, if
\[ P(x)=x^2-5x+6, \]then
\[ P(2)=4-10+6=0 \]and
\[ P(3)=9-15+6=0. \]Therefore,
\[ \boxed{2\text{ and }3\text{ are zeros of }P}. \]Root of a Polynomial Equation
A root is a solution of the polynomial equation
\[ P(x)=0. \]For
\[ x^2-5x+6=0, \]the roots are
\[ \boxed{x=2,\ 3}. \]In polynomial work, the terms zero and root often refer to the same numerical values, but the language emphasizes different viewpoints:
- zero emphasizes the function \(P(c)=0\);
- root emphasizes solving the equation \(P(x)=0\).
Factor Language
By the Factor Theorem,
\[ P(c)=0 \]if and only if
\[ x-c \]is a factor of \(P(x)\).
Since
\[ x^2-5x+6=(x-2)(x-3), \]the factors directly reveal the roots
\[ \boxed{2,\ 3}. \]Graph Language
If \(c\) is a real zero, then
\[ P(c)=0. \]Therefore, the graph contains the point
\[ (c,0). \]This is an \(x\)-intercept.
Thus a real root can be recognized symbolically, numerically, or graphically.
Four Equivalent Views
| View | Statement |
|---|---|
| Function | \(P(c)=0\) |
| Equation | \(c\) solves \(P(x)=0\) |
| Factorization | \(x-c\) is a factor |
| Graph | \((c,0)\) is an \(x\)-intercept, if \(c\) is real |
From a Root to a Factor
Suppose
\[ x=4 \]is a root.
Then the corresponding factor is
\[ \boxed{x-4}. \]The sign changes because the factor must become zero when \(x=4\):
\[ 4-4=0. \]From a Factor to a Root
Suppose
\[ x+7 \]is a factor.
Set it equal to zero:
\[ x+7=0. \]Therefore,
\[ \boxed{x=-7}. \]So the factor \(x+7\) corresponds to the root \(-7\).
Factored Form Reveals Zeros
Consider
\[ P(x)=3(x+2)(x-1)(x-5). \]The zeros are immediately visible:
\[ \boxed{x=-2,\ 1,\ 5}. \]The leading coefficient \(3\) changes the vertical scale of the graph but does not change the locations of these zeros.
What Is Multiplicity?
A root has multiplicity equal to the number of times its corresponding linear factor occurs.
If
\[ P(x)=(x-3)^2(x+1), \]then:
- \(x=3\) has multiplicity \(2\);
- \(x=-1\) has multiplicity \(1\).
Multiplicity 1
If a factor appears once,
\[ (x-c)^1, \]then \(c\) is a root of multiplicity \(1\).
This is sometimes called a simple root.
Multiplicity 2
If
\[ (x-c)^2 \]divides the polynomial but
\[ (x-c)^3 \]does not, then \(c\) has multiplicity \(2\).
Such a root is repeated twice.
Higher Multiplicity
If
\[ P(x)=(x-c)^mQ(x) \]and
\[ Q(c)\neq0, \]then \(c\) is a root of multiplicity
\[ \boxed{m}. \]The condition \(Q(c)\neq0\) ensures that no additional factor of \(x-c\) remains hidden inside \(Q(x)\).
Example: Read Multiplicity from Factored Form
Consider
\[ P(x)=2(x+4)^3(x-1)^2(x-5). \]The roots and multiplicities are:
- \(x=-4\), multiplicity \(3\);
- \(x=1\), multiplicity \(2\);
- \(x=5\), multiplicity \(1\).
Total Multiplicity and Degree
In a completely factored polynomial over the complex numbers, the sum of the root multiplicities equals the degree.
For
\[ P(x)=2(x+4)^3(x-1)^2(x-5), \]the total multiplicity is
\[ 3+2+1=6. \]Therefore, the polynomial has degree
\[ \boxed{6}. \]Distinct Roots Versus Roots Counted with Multiplicity
The polynomial
\[ (x-2)^3(x+5)^2 \]has only two distinct roots:
\[ 2,\ -5. \]But counted with multiplicity, it has
\[ 3+2=5 \]roots.
This distinction becomes important when relating roots to polynomial degree.
Multiplicity and Graph Behavior
The parity of a real root's multiplicity affects how the graph behaves at the corresponding \(x\)-intercept.
- Odd multiplicity: the graph crosses the \(x\)-axis.
- Even multiplicity: the graph touches the axis and turns back.
Higher multiplicity also tends to make the graph flatter near the zero.
Why Odd Multiplicity Crosses
Near a root \(x=c\), the factor
\[ (x-c)^m \]changes sign when \(m\) is odd.
For example,
\[ (x-c)^3 \]is negative on one side of \(c\) and positive on the other.
This sign change forces the graph through the \(x\)-axis.
Why Even Multiplicity Touches and Turns
If \(m\) is even, then
\[ (x-c)^m \]has the same sign on both sides of \(c\).
For example,
\[ (x-c)^2\ge0. \]The graph reaches zero but does not change sign because of that factor.
It therefore touches the axis and turns back.
Multiplicity 1 Graph Behavior
A simple root typically produces an ordinary crossing.
For
\[ P(x)=x-2, \]the root \(x=2\) has multiplicity \(1\).
The graph crosses the axis at
\[ \boxed{(2,0)}. \]Multiplicity 2 Graph Behavior
Consider
\[ P(x)=(x-2)^2. \]The root \(2\) has multiplicity \(2\).
The graph touches the axis at
\[ (2,0) \]and turns back without crossing.
Multiplicity 3 Graph Behavior
Consider
\[ P(x)=(x-2)^3. \]The root \(2\) has multiplicity \(3\).
The graph crosses the \(x\)-axis because the multiplicity is odd, but the crossing is flatter than for multiplicity \(1\).
Multiplicity 4 Graph Behavior
Consider
\[ P(x)=(x-2)^4. \]The root \(2\) has multiplicity \(4\).
The graph touches and turns because the multiplicity is even, and the graph is flatter near the intercept than it is for multiplicity \(2\).
Local Power-Function Behavior
Near a root \(x=c\), a polynomial containing
\[ (x-c)^m \]behaves locally like a shifted power function.
The same even-versus-odd power patterns studied earlier now explain intercept behavior:
- odd \(m\): crossing;
- even \(m\): touching and turning.
Example: Analyze All Intercepts
Consider
\[ P(x)=(x+3)^2(x-1)^3(x-5). \]At \(x=-3\), multiplicity \(2\) is even, so the graph touches and turns.
At \(x=1\), multiplicity \(3\) is odd, so the graph crosses with a flatter transition.
At \(x=5\), multiplicity \(1\) is odd, so the graph crosses normally.
Repeated Roots in Equations
Solve
\[ (x-4)^2(x+1)=0. \]The solutions are
\[ x=4 \]and
\[ x=-1. \]The solution \(x=4\) occurs twice in the factorization, so it has multiplicity \(2\).
The solution set itself is
\[ \boxed{\{-1,4\}}, \]but multiplicity records the repeated algebraic structure.
Multiplicity Is Not the Same as Writing the Root Repeatedly
It is usually better to write
\[ x=4\text{ with multiplicity }2 \]than to list
\[ x=4,\ 4. \]The first notation clearly distinguishes the root value from the number of times its factor occurs.
Repeated Roots from Synthetic Division
Suppose synthetic division shows that \(x-2\) is a factor of \(P(x)\).
Divide by \(x-2\).
If the quotient also has \(x-2\) as a factor, then \(2\) has multiplicity at least \(2\).
Continue dividing by the same factor until the remainder is no longer zero.
The number of successful divisions gives the multiplicity.
Example: Detect Multiplicity by Repeated Division
Let
\[ P(x)=x^3-6x^2+12x-8. \]Test \(x=2\):
\[ P(2)=0. \]Divide by \(x-2\):
\[ x^2-4x+4. \]That quotient is
\[ (x-2)^2. \]Therefore,
\[ \boxed{P(x)=(x-2)^3}. \]The root \(2\) has multiplicity \(3\).
Complex Roots Also Have Multiplicity
Multiplicity is not limited to real roots.
For example,
\[ P(x)=(x-i)^2(x+i)^2 \]has roots
\[ i \]and
\[ -i, \]each with multiplicity \(2\).
These roots do not appear as real \(x\)-intercepts because they are not real numbers.
Conjugate Multiplicities for Real Coefficients
If a polynomial has real coefficients and a nonreal root
\[ a+bi, \]then
\[ a-bi \]is also a root.
In fact, the conjugate roots occur with the same multiplicity.
This preserves real coefficients when the corresponding factors are multiplied.
Degree Places a Limit on Distinct Zeros
A nonzero polynomial of degree \(n\) can have at most
\[ \boxed{n} \]distinct zeros.
For example, a cubic polynomial cannot have four distinct roots.
Repeated roots may reduce the number of distinct roots below the degree.
Degree and Root Count over the Complex Numbers
A degree-\(n\) polynomial with complex coefficients has exactly \(n\) complex roots when roots are counted with multiplicity.
This is a consequence of the Fundamental Theorem of Algebra.
Thus a degree-\(5\) polynomial has five roots counted with multiplicity, even if some roots repeat or are nonreal.
Example: Degree and Multiplicity
Consider
\[ P(x)=(x-1)^2(x+4)^3. \]The polynomial has degree
\[ 2+3=5. \]It has two distinct roots:
\[ 1,\ -4. \]Counted with multiplicity, it has
\[ 2+3=5 \]roots, matching the degree.
Building a Polynomial from Given Roots
If the roots are
\[ r_1,\ r_2,\ldots,r_n, \]then a polynomial with those roots can be built from
\[ (x-r_1)(x-r_2)\cdots(x-r_n). \]A nonzero leading constant may also be included:
\[ \boxed{ P(x)=a(x-r_1)(x-r_2)\cdots(x-r_n) }. \]Build a Polynomial with Multiplicity
Construct a monic polynomial with:
- root \(2\) of multiplicity \(3\);
- root \(-1\) of multiplicity \(2\).
The factors are
\[ (x-2)^3 \]and
\[ (x+1)^2. \]Therefore,
\[ \boxed{P(x)=(x-2)^3(x+1)^2}. \]Minimum Degree from Root Information
If a polynomial must have roots with specified multiplicities, those multiplicities determine a minimum degree.
Suppose a polynomial has:
- root \(1\) with multiplicity \(2\);
- root \(4\) with multiplicity \(1\);
- root \(-3\) with multiplicity \(3\).
The minimum possible degree is
\[ \boxed{2+1+3=6}. \]Zeros from Expanded Form
Expanded form usually does not reveal multiplicity immediately.
Consider
\[ P(x)=x^3-3x^2+4. \]To identify zeros and multiplicities, we may need to factor, test roots, or use synthetic division.
Factored form is the most transparent representation for this purpose.
Example: Factor and Read Multiplicity
Factor
\[ x^3-3x^2+4. \]Test \(x=2\):
\[ 8-12+4=0. \]Divide by \(x-2\):
\[ x^2-x-2. \]Factor:
\[ x^2-x-2=(x-2)(x+1). \]Therefore,
\[ \boxed{P(x)=(x-2)^2(x+1)}. \]So \(2\) has multiplicity \(2\), while \(-1\) has multiplicity \(1\).
Sign Changes Near a Root
An odd-multiplicity factor changes sign as \(x\) moves through the root.
An even-multiplicity factor does not.
Therefore, multiplicity can often be inferred from a graph:
- crossing suggests odd multiplicity;
- touching and turning suggests even multiplicity.
The graph alone may not always reveal the exact multiplicity, especially at high scale, but it reveals the parity of the multiplicity locally.
Multiplicity and Flatness
Compare
\[ y=x, \qquad y=x^3, \qquad y=x^5. \]All cross the origin because their exponents are odd.
But the higher powers flatten increasingly near \(x=0\).
The same local behavior appears near a polynomial root of multiplicity \(3\), \(5\), and so on.
Common Error: Confusing a Root with Its Factor
If the root is
\[ 5, \]the factor is
\[ \boxed{x-5}, \]not \(x+5\).
Conversely, the factor \(x+5\) corresponds to root \(-5\).
Common Error: Ignoring Multiplicity
The factorization
\[ (x-2)^3(x+1) \]contains more information than merely saying the roots are \(2\) and \(-1\).
The root \(2\) occurs three times and has multiplicity \(3\).
That repetition affects both algebraic root count and graph behavior.
Common Error: Assuming Every Intercept Is a Simple Crossing
A graph does not always cross the axis at a real zero.
If the root has even multiplicity, the graph touches the axis and turns back.
The factor exponent must be considered.
Common Error: Counting Distinct Roots as the Degree
The polynomial
\[ (x-1)^4 \]has degree \(4\), but only one distinct root:
\[ x=1. \]Counted with multiplicity, that single root contributes four times.
Common Error: Assuming Complex Roots Give Real Intercepts
The roots
\[ 2\pm3i \]are valid roots of a polynomial equation, but they do not correspond to points on the real \(x\)-axis.
Only real roots produce real \(x\)-intercepts.
A Reliable Procedure for Reading Factored Form
- Identify every linear factor \(x-c\).
- Set each factor equal to zero.
- Record the corresponding root \(c\).
- Read the exponent on each factor as the multiplicity.
- Use multiplicity parity to predict crossing or touching for real roots.
- Check that the total multiplicity is consistent with the polynomial degree.
A Reliable Procedure for Finding Multiplicity from Expanded Form
- Find or test a root \(c\).
- Use synthetic or polynomial division by \(x-c\).
- If the remainder is zero, divide again by \(x-c\).
- Continue until the remainder is nonzero.
- Count the number of successful divisions.
- That count is the multiplicity of \(c\).
Practice 1: Read Roots and Multiplicities
For
\[ P(x)=4(x-2)^3(x+5)^2(x-1), \]list all roots and their multiplicities.
Practice 2: Predict Graph Behavior
For the polynomial in Practice 1, state whether the graph crosses or touches the \(x\)-axis at each real zero.
Practice 3: Build a Polynomial
Write a monic polynomial in factored form with:
- root \(3\) of multiplicity \(2\);
- root \(-4\) of multiplicity \(1\).
Practice 4: Degree from Multiplicity
What is the degree of
\[ P(x)=7(x+1)^4(x-2)^3? \]Practice 5: Distinct Roots Versus Total Multiplicity
For
\[ P(x)=(x-6)^2(x+3)^5, \]state:
- the number of distinct roots;
- the number of roots counted with multiplicity.
Practice 6: Find Multiplicity by Factoring
Factor and identify multiplicities:
\[ x^3-3x^2+4. \]Practice 7: Factor from Root Information
If \(x=-2\) is a root of multiplicity \(3\), what factor must divide the polynomial?
Practice 8: Graph Interpretation
A graph touches the \(x\)-axis at \(x=4\) and turns back.
What can you conclude about the multiplicity parity of the root \(4\)?
Practice 9: Complex Roots
A real-coefficient polynomial has root
\[ 3+2i \]of multiplicity \(2\).
What other root must occur, and with what multiplicity?
Practice 10: Minimum Degree
Find the minimum possible degree of a polynomial having:
- root \(1\) of multiplicity \(2\);
- root \(-3\) of multiplicity \(4\);
- root \(5\) of multiplicity \(1\).
Practice Answers
Practice 1
- \(x=2\), multiplicity \(\boxed{3}\);
- \(x=-5\), multiplicity \(\boxed{2}\);
- \(x=1\), multiplicity \(\boxed{1}\).
Practice 2
- At \(x=2\): odd multiplicity, so the graph crosses.
- At \(x=-5\): even multiplicity, so the graph touches and turns.
- At \(x=1\): odd multiplicity, so the graph crosses.
Practice 3
\[ \boxed{P(x)=(x-3)^2(x+4)}. \]Practice 4
\[ 4+3=\boxed{7}. \]Practice 5
Distinct roots:
\[ \boxed{2}. \]Counted with multiplicity:
\[ 2+5=\boxed{7}. \]Practice 6
As shown earlier,
\[ x^3-3x^2+4=(x-2)^2(x+1). \]Therefore:
- \(x=2\), multiplicity \(\boxed{2}\);
- \(x=-1\), multiplicity \(\boxed{1}\).
Practice 7
Root \(-2\) corresponds to factor
\[ x+2. \]Multiplicity \(3\) gives
\[ \boxed{(x+2)^3}. \]Practice 8
Touching and turning indicates
\[ \boxed{\text{even multiplicity}}. \]Practice 9
The conjugate root is
\[ \boxed{3-2i}, \]also with multiplicity
\[ \boxed{2}. \]Practice 10
\[ 2+4+1=\boxed{7}. \]Therefore, the minimum possible degree is \(7\).
What to Remember
For a polynomial \(P(x)\),
\[ \boxed{ P(c)=0 \iff c\text{ is a zero} \iff c\text{ is a root} \iff x-c\text{ is a factor} }. \]If \(c\) is real, then \((c,0)\) is also an \(x\)-intercept.
If
\[ P(x)=(x-c)^mQ(x) \]with \(Q(c)\neq0\), then \(c\) has multiplicity
\[ \boxed{m}. \]For real roots:
- odd multiplicity usually means the graph crosses the \(x\)-axis;
- even multiplicity means the graph touches and turns;
- larger multiplicity produces flatter local behavior near the zero.
A degree-\(n\) polynomial has exactly \(n\) complex roots counted with multiplicity.
Where We Go Next
Zeros and multiplicity describe what happens where a polynomial meets the horizontal axis. To understand the entire graph, we also need to know what happens far to the left and right and how many times the graph can change direction.
The next lesson, End Behavior and Turning Points, connects degree and leading coefficient to global polynomial shape and develops the maximum number of turning points a polynomial can have.