AP PRECALCULUS — UNIT 1A · POLYNOMIAL & RATIONAL FUNCTIONS
1.5A — Polynomial Functions and Complex Zeros
Notes — Zeros, Multiplicity, and the Fundamental Theorem
💡 Learning Objectives (1.5.A.1 – 1.5.A.6) By the end of this lesson you will be able to:
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1. Zeros and the Factor Theorem
💡 Zero of a polynomial A complex number a is called a zero of the polynomial p(x) (or a root of the equation p(x) = 0) if substituting x = a makes the polynomial evaluate to 0: p(a) = 0. If a is a real number, this means the graph of y = p(x) crosses or touches the x-axis at the point (a, 0). |
💡 The Factor Theorem For a real number a: (x − a) is a factor of p(x) ⟺ a is a zero of p(x). In other words, factoring out (x − a) and finding zeros are two views of the same fact. |
📘 Example — Using the factor theorem Verify that x = 2 is a zero of p(x) = x³ − 4x² + x + 6. p(2) = 8 − 16 + 2 + 6 = 0 ✓. Therefore (x − 2) is a factor. Divide to obtain p(x) = (x − 2)(x² − 2x − 3) = (x − 2)(x − 3)(x + 1). All three real zeros: x = 2, x = 3, x = −1. |
2. Multiplicity of a Zero
💡 Multiplicity, defined If the linear factor (x − a) appears exactly n times in the factored form of p(x), then a is a zero of multiplicity n. We say the zero ‘has multiplicity n.’ Example: p(x) = (x − 1)²(x + 3)(x − 5)³ has three distinct real zeros: 1 (multiplicity 2), −3 (multiplicity 1), and 5 (multiplicity 3). |
2.1 What multiplicity looks like on a graph
Behavior at a real zero depends on its multiplicity. Odd multiplicity ⇒ the curve crosses the x-axis. Even multiplicity ⇒ the curve only touches the x-axis and bounces back the way it came.
Multiplicity | Graph behaviour at the zero | Sign change? |
|---|---|---|
1 (odd) | Crosses straight through the x-axis | Yes |
2 (even) | Touches the x-axis and turns away (tangent) | No |
3 (odd) | Crosses, with a brief flattening | Yes |
4 (even) | Touches and bounces, very flat | No |
odd ≥ 5 | Crosses with strong flattening | Yes |
even ≥ 6 | Touches and bounces, extremely flat | No |
⚠️ Key observation At a real zero of even multiplicity, the polynomial doesn't change sign — values just to the left and just to the right of the zero have the same sign. This matters for solving polynomial inequalities. |
3. The Fundamental Theorem of Algebra
💡 FTA — Counting all the zeros Every nonconstant polynomial of degree n with real (or complex) coefficients has exactly n complex zeros, when zeros are counted according to their multiplicity. Some of those zeros may be real numbers; others may be non-real complex numbers of the form a + bi. |
3.1 Conjugate pairs
💡 Complex zeros come in pairs If a polynomial has real coefficients and (a + bi) is a non-real zero (so b ≠ 0), then its complex conjugate (a − bi) is also a zero. This means non-real zeros always show up in conjugate pairs in real polynomials. Counting them up, the number of non-real zeros must be even. |
📘 Example — Counting all the zeros A degree-5 polynomial p with real coefficients has zeros 2 (multiplicity 2), 1 + i, and 1 − i. Counts: 2 (twice) + (1+i) + (1−i) = 4 zeros so far. The 5th zero must be a real number — say x = −3 — to bring the total (with multiplicities) to 5. Possible factored form: p(x) = a(x − 2)²(x + 3)(x − (1+i))(x − (1−i)) = a(x − 2)²(x + 3)(x² − 2x + 2). |
4. Real Zeros and Polynomial Inequalities
The real zeros of p divide the real number line into intervals. Within each such interval, p(x) keeps a constant sign (either always positive or always negative). To solve p(x) > 0 or p(x) < 0:
- Step 1: find all real zeros (with multiplicity).
- Step 2: place them on a number line.
- Step 3: test one value from each interval to determine the sign of p there.
- Step 4: remember: at a real zero of even multiplicity, the sign does not flip.
📘 Example — Solving a polynomial inequality Solve (x + 2)(x − 1)²(x − 4) ≤ 0. Real zeros: −2, 1 (mult 2), 4. Sign check on intervals (−∞, −2), (−2, 1), (1, 4), (4, ∞): + , − , − , + . (The sign does NOT change at x = 1 because that zero has even multiplicity.) So (x + 2)(x − 1)²(x − 4) ≤ 0 on [−2, 4] (closed because of the ≤). |
5. Determining the Degree from Data
If a table of values is generated by a polynomial sampled at equally spaced inputs, the degree of that polynomial is the order at which the successive differences first become constant.
Repeated differences uncover the polynomial's degree. Here the third differences are constant, so the underlying function is a cubic.
📘 Example — Inferring degree from a table Inputs at x = 0, 1, 2, 3, 4, 5 give outputs: 0, 1, 16, 81, 256, 625. 1st diffs: 1, 15, 65, 175, 369. 2nd diffs: 14, 50, 110, 194. 3rd diffs: 36, 60, 84. 4th diffs: 24, 24. ← constant ⇒ degree 4. (Indeed, the formula is p(x) = x⁴.) |
6. Summary
- a is a zero of p iff (x − a) is a linear factor of p iff p(a) = 0
- Multiplicity n means (x − a) appears n times; even multiplicity ⇒ touch-and-bounce, odd ⇒ cross
- Fundamental Theorem of Algebra: degree n has exactly n complex zeros (with multiplicity)
- Non-real zeros of real polynomials come in complex conjugate pairs
- Real zeros with their multiplicities tell you exactly where the sign of p flips
- The degree of a polynomial fit to equally spaced data equals the order at which successive differences first become constant