Unit 1 · Topic 1.5
Polynomial Functions and Complex Zeros
Connect zeros, factors, multiplicity, graph behavior, conjugate pairs, finite differences, and symmetry to build a complete description of a polynomial function.
1. Zeros, Roots, Factors, and Intercepts
A number \(z\) is a zero of \(P\) when \(P(z)=0\). It is also a root, or solution, of the equation \(P(x)=0\).
\[P(a)=0\quad\Longleftrightarrow\quad (x-a)\text{ is a factor of }P(x).\]
When \(a\) is real, the zero produces the x-intercept \((a,0)\). A nonreal zero solves the equation but cannot appear on a real coordinate graph.
2. Complex Numbers and Conjugates
A complex number has the form \(a+bi\), where \(i^2=-1\). Its complex conjugate is \(a-bi\).
\[(a+bi)(a-bi)=a^2+b^2.\]
The imaginary terms cancel in the product. This explains why nonreal zeros of a polynomial with real coefficients occur in conjugate pairs.
\[a+bi\text{ is a zero}\quad\Longrightarrow\quad a-bi\text{ is also a zero}.\]
3. Degree Counts Zeros with Multiplicity
A degree-\(n\) polynomial has exactly \(n\) complex zeros when every repetition is counted. Real numbers are included within the complex number system.
\[P(x)=A(x-z_1)(x-z_2)\cdots(x-z_n),\qquad A\ne0.\]
| Polynomial | Degree | Zeros counted with multiplicity |
|---|
| \((x-4)^3\) | 3 | \(4,4,4\) |
| \((x+1)(x^2+9)\) | 3 | \(-1,3i,-3i\) |
| \((x-2)^2(x+5)^2\) | 4 | \(2,2,-5,-5\) |
Multiplicity is part of the count. Three distinct zeros do not necessarily mean degree 3 if one or more zeros repeat.
4. Multiplicity Predicts Local Graph Behavior
If \((x-a)^m\) is a factor, then \(a\) is a zero with multiplicity \(m\).
| Multiplicity | Signs near \(x=a\) | Graph at the x-axis |
|---|
| Even | Same sign on both sides | Touches the axis and turns |
| Odd | Opposite signs on the two sides | Crosses the axis |
A larger multiplicity can make the graph look flatter near the intercept, but parity determines whether the sign changes.
5. Worked Factored-Form Example
Analyze \(P(x)=(x-2)^2(x+1)(x^2+4)\).
- Add factor degrees: \(2+1+2=5\), so \(P\) has degree 5.
- Real zeros are \(x=2\) with multiplicity 2 and \(x=-1\) with multiplicity 1.
- From \(x^2+4=0\), the nonreal zeros are \(x=2i\) and \(x=-2i\).
- The graph touches the x-axis at \(x=2\) and crosses it at \(x=-1\).
All five zeros are accounted for: \(2,2,-1,2i,-2i\).
6. Find Missing Zeros from a Conjugate Pair
Suppose a degree-4 polynomial with real coefficients has zeros \(5\), \(-2\), and \(1+3i\). The conjugate \(1-3i\) must also be a zero.
\[P(x)=A(x-5)(x+2)(x-(1+3i))(x-(1-3i)).\]
Pair the conjugate factors first:
\[(x-(1+3i))(x-(1-3i))=(x-1)^2+9=x^2-2x+10.\]
This quadratic factor has real coefficients, as the original polynomial requires.
7. Construct a Polynomial from Zeros and a Point
Find a degree-3 polynomial with real coefficients whose zeros are \(3\) and \(-1+2i\), and whose graph contains \((0,30)\).
- Include the conjugate zero \(-1-2i\).
- Write \(P(x)=A(x-3)(x+1-2i)(x+1+2i)\).
- Multiply the conjugates: \((x+1)^2+4=x^2+2x+5\).
- Use \(P(0)=30\): \(30=A(-3)(5)\), so \(A=-2\).
\[P(x)=-2(x-3)(x^2+2x+5)=-2x^3+2x^2+2x+30.\]
Check the result by confirming its degree, real coefficients, stated zeros, and known point.
8. Real Zeros Organize Polynomial Inequalities
Real zeros divide the number line into intervals where a polynomial keeps one sign. Consider
\[H(x)=(x+2)(x-1)^2(x-4).\]
| Interval or zero | Sign of \(H(x)\) | Reason |
|---|
| \(( -\infty,-2)\) | Positive | Two linear factors are negative. |
| \(x=-2\) | Zero | Odd multiplicity; the sign changes. |
| \((-2,1)\) | Negative | One linear factor is negative. |
| \(x=1\) | Zero | Even multiplicity; the sign does not change. |
| \((1,4)\) | Negative | The squared factor stays positive away from 1. |
| \(x=4\) | Zero | Odd multiplicity; the sign changes. |
| \((4,\infty)\) | Positive | Every factor is positive. |
Therefore \(H(x)\ge0\) on \(( -\infty,-2]\cup\{1\}\cup[4,\infty)\). Include a zero for \(\le\) or \(\ge\); exclude it for a strict inequality.
9. Find Degree from Successive Differences
For equally spaced inputs, the least order of constant nonzero differences identifies the polynomial degree. Let \(F(x)=x^3-x\).
| Difference order | Values from left to right | Conclusion |
|---|
| Outputs at \(x=0,1,2,3,4\) | \(0,0,6,24,60\) | Start with equally spaced inputs. |
|---|
| First differences | \(0,6,18,36\) | Not constant |
|---|
| Second differences | \(6,12,18\) | Not constant |
|---|
| Third differences | \(6,6\) | Constant |
|---|
The first and second differences vary, while the third differences are constant. The least such order is 3, so the data come from a cubic polynomial.
Important: this test requires equal input spacing. Constant third differences do not identify every coefficient; they identify the degree.
10. Even, Odd, or Neither
| Classification | Analytical test | Graph symmetry | Polynomial pattern |
|---|
| Even | \(P(-x)=P(x)\) | Symmetric about the y-axis | Only even powers may have nonzero coefficients. |
| Odd | \(P(-x)=-P(x)\) | Symmetric about the origin | Only odd powers may have nonzero coefficients; constant term is 0. |
| Neither | Neither identity holds | Neither required symmetry | Usually contains a mix of even and odd powers. |
For \(E(x)=4x^4-3x^2+8\), replacing \(x\) by \(-x\) leaves the formula unchanged, so \(E\) is even. For \(O(x)=2x^5-7x\), substitution gives \(-O(x)\), so \(O\) is odd.
11. Technology Workflow
- Factor analytically when a useful factorization is available.
- Use a graph to estimate real zeros and observe crossing or touching behavior.
- Use a table near each real zero to verify whether the sign changes.
- Use a polynomial-solver tool to estimate remaining real or nonreal zeros.
- Count every zero with multiplicity and compare the total with the degree.
- Verify approximate zeros by substituting them into the polynomial.
A graph cannot display nonreal zeros, and a poor viewing window can hide real zeros. Combine analytical and technological evidence.
12. Common Errors
- Counting distinct zeros instead of counting multiplicity.
- Forgetting the conjugate of a nonreal zero when coefficients are real.
- Assuming every complex zero is nonreal; every real number is also complex.
- Drawing nonreal zeros as x-intercepts.
- Assuming the sign changes at an even-multiplicity zero.
- Using successive differences when input intervals are not equal.
- Calling every even-degree polynomial an even function.
- Solving \(P(x)>0\) from the zeros without testing interval signs.
13. AP Reasoning Focus
Zero-to-graph pattern: “Because \((x-a)^m\) is a factor, \(a\) is a zero of multiplicity \(m\). Since \(m\) is even/odd, the graph keeps/changes sign at \(x=a\).”
Conjugate pattern: “The coefficients are real, so the nonreal zero \(a+bi\) requires the conjugate zero \(a-bi\). Their factors multiply to a real quadratic.”
Symmetry pattern: Substitute \(-x\), simplify fully, and compare the result with both \(P(x)\) and \(-P(x)\).