Polynomial Functions and Complex Zeros
Introduced| English | Meaning |
|---|---|
| degree/dɪˈɡriː/ | The highest power of a variable in a single-variable polynomial with non-zero coefficients. |
| real zeros/rɪəl ˈzɪərəʊz/ | The x-intercepts, with even/odd multiplicity deciding touch vs cross. |
| multiplicity/ˌmʌltɪˈplɪsɪti/ | The number of times a root occurs as a repeated factor of a polynomial. |
| factored form/ˈfæktəd fɔːm/ | factored form |
| complex zeros/ˈkɒmpleks ˈzɪərəʊz/ | complex zeros |
| conjugate pairs/ˈkɒndʒuːɡeɪt peəz/ | conjugate pairs |
Where did the other roots go?
- The graph of $y = x^2 + 1$ is a parabola floating entirely above the x-axis.
- It never crosses, so it seems to have no solutions to $x^2 + 1 = 0$.
- Yet algebra insists a degree-2 equation should have two roots.
- The missing roots are hiding in the complex numbers.
Real zeros and multiplicity
- The real zeros 实零点 of a polynomial are the x-values where its graph crosses or touches the x-axis.
- Read them straight from the factored form 因式分解形式: $(x-2)(x+3)$ has zeros at $2$ and $-3$.
- A repeated factor like $(x-2)^2$ gives a zero of multiplicity 重数 two.
- Even multiplicity → the graph touches and bounces; odd multiplicity → it crosses.
A zero where the graph just touches the x-axis (instead of crossing) has an even ____.
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Even multiplicity (like $(x-3)^2$) makes the curve bounce off the axis; odd multiplicity makes it cross.
When the graph misses the axis
- Some polynomials never reach the x-axis, so they have fewer real zeros than their degree.
- The zeros that are "missing" are complex zeros 复零点 — values built with $i = \sqrt{-1}$.
- They are just as real as solutions; they simply do not appear as x-intercepts.
A parabola that never meets the x-axis has how many real zeros?
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No x-intercepts means no real zeros. But it still has two zeros — they are complex.
Complex zeros travel in pairs
- If a polynomial has real coefficients and $a + bi$ is a zero, then $a - bi$ is a zero too.
- These mirror-image partners are called conjugate pairs 共轭对.
- That is why complex zeros always arrive two at a time — never alone.
- So a real polynomial can have $0, 2, 4, \dots$ complex zeros, but never an odd number.
A complex zero and its mirror partner
Plot a zero a + bi. For a real polynomial its conjugate a − bi (the mirror image across the real axis) must be a zero too.
For a polynomial with real coefficients, a complex zero $a + bi$ can appear without $a - bi$.
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False. With real coefficients, complex zeros come in conjugate pairs: if $a + bi$ is a zero, so is $a - bi$.
Counting every zero
- A polynomial of degree $n$ has exactly $n$ zeros, once you count real and complex zeros with multiplicity.
- This is the Fundamental Theorem of Algebra, and it never fails.
- So a cubic always has $3$ zeros; a quartic always has $4$.
- If you can see fewer x-intercepts than the degree 次数, the rest are complex.
Counted with multiplicity, how many zeros does a degree 4 polynomial have in total?
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A degree-$n$ polynomial has exactly $n$ zeros in total, real and complex counted with multiplicity.
A cubic (degree 3) has exactly 1 real zero. How many complex (non-real) zeros must it have?
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Total zeros $= 3$. With $1$ real, the other $2$ are a complex conjugate pair.
Select all true statements.
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A missing x-intercept does not reduce the zero count — the missing zeros are complex. The others are all true.
A "missing" x-intercept does not mean the polynomial has fewer zeros. The zero count always equals the degree — the unseen zeros are simply complex. Do not report a quartic as having "two zeros" just because you see two x-intercepts.
Solve $x^2 + 1 = 0$.
- Rearranged: $x^2 = -1$, so $x = \pm\sqrt{-1} = \pm i$.
- The two zeros are $i$ and $-i$ — a conjugate pair, exactly as promised.
- The parabola $y = x^2 + 1$ floats above the axis because neither zero is real.
The real zeros are the x-intercepts, with even/odd multiplicity deciding touch vs cross. Any remaining zeros are complex zeros, which occur in conjugate pairs. Counted with multiplicity, the total number of zeros always equals the degree.
Write from 5 zeros
- Count zeros of a degree-5 polynomial with multiplicity.
- Complex zeros are included.
Counted with multiplicity, how many zeros does a polynomial of degree 5 have?
The degree is the total number of zeros.
How many zeros does degree 5 have, with multiplicity?
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Degree and total zero count match.