Problem
The phrase “how to find all real zeros of a polynomial” means finding every real number \(r\) such that \(P(r)=0\). Consider \[ P(x)=x^4-4\cdot x^3+3\cdot x^2+4\cdot x-4. \] Determine all real zeros and the multiplicity of each.
Meaning of a real zero and multiplicity
A real zero is a real value \(r\) where \(P(r)=0\). On the graph \(y=P(x)\), real zeros are the x-intercepts. If a factor \((x-r)^k\) appears in the factorization, then \(r\) is a zero of multiplicity \(k\).
Odd multiplicity: the graph typically crosses the x-axis at \(r\). Even multiplicity: the graph typically touches the x-axis and turns at \(r\).
Step 1: List rational candidates (Rational Root Theorem)
Since the leading coefficient is \(1\), any rational real zero must be an integer divisor of the constant term \(-4\). Therefore, the candidates are: \[ \pm 1,\ \pm 2,\ \pm 4. \]
Step 2: Test candidates
Evaluate \(P(x)\) at candidate integers.
| Candidate \(x\) | \(P(x)\) | Result |
|---|---|---|
| \(1\) | \(1-4+3+4-4=0\) | \(x=1\) is a real zero |
| \(-1\) | \(1+4+3-4-4=0\) | \(x=-1\) is a real zero |
| \(2\) | \(16-32+12+8-4=0\) | \(x=2\) is a real zero |
| \(4\) | \(256-256+48+16-4=60\) | Not a zero |
Step 3: Synthetic division by \((x-1)\)
Coefficients of \(P(x)\) are \(1,\ -4,\ 3,\ 4,\ -4\).
| 1 | -4 | 3 | 4 | -4 | |
|---|---|---|---|---|---|
| Bring down | 1 | ||||
| Multiply by 1 | 1 | -3 | 0 | 4 | |
| Add | 1 | -3 | 0 | 4 | 0 |
The remainder is \(0\), so \[ P(x)=(x-1)\cdot\left(x^3-3\cdot x^2+4\right). \]
Step 4: Synthetic division by \((x-2)\) and factoring
Let \(Q(x)=x^3-3\cdot x^2+4\). Divide \(Q(x)\) by \((x-2)\):
| 1 | -3 | 0 | 4 | |
|---|---|---|---|---|
| Bring down | 1 | |||
| Multiply by 2 | 2 | -2 | -4 | |
| Add | 1 | -1 | -2 | 0 |
Thus \[ Q(x)=(x-2)\cdot(x^2-x-2). \] Factor the quadratic: \[ x^2-x-2=(x-2)\cdot(x+1). \] Therefore, \[ Q(x)=(x-2)^2\cdot(x+1). \]
Step 5: All real zeros with multiplicity
Combine factors: \[ P(x)=(x-1)\cdot(x+1)\cdot(x-2)^2. \]
Real zeros: \(x=-1,\ 1,\ 2\)
Multiplicities: \(-1\) has multiplicity \(1\), \(1\) has multiplicity \(1\), and \(2\) has multiplicity \(2\).
Summary table
| Zero \(r\) | Factor | Multiplicity | Typical x-axis behavior |
|---|---|---|---|
| \(-1\) | \((x+1)\) | 1 | Crosses |
| \(1\) | \((x-1)\) | 1 | Crosses |
| \(2\) | \((x-2)^2\) | 2 | Touches and turns |
Visualization: number line of real zeros and multiplicity
Common pitfalls
- Stopping after one root: after finding a zero, divide to reduce the degree and continue until factors are fully simplified.
- Missing repeated roots: if division produces a factor \((x-r)^2\), the zero \(r\) must be counted twice by multiplicity.
- Assuming all roots are rational: the Rational Root Theorem lists only possible rational zeros; irrational real zeros require other methods (factoring patterns, completing the square, or numerical techniques).