Polynomials

4A. Complex Numbers

1) Real and Imaginary Parts

Real Part

If

$$z=a+bi,$$

then

$$\operatorname{Re}z=a.$$

Meaning: The real part is the real-number component of a complex number.

Imaginary Part

If

$$z=a+bi,$$

then

$$\operatorname{Im}z=b.$$

Thus,

$$z=\operatorname{Re}z+(\operatorname{Im}z)i.$$

Meaning: The imaginary part is the coefficient of $i$.

2) Complex Conjugate

Complex Conjugate

For $z\in\mathbb{C}$,

$$\overline{z} = \operatorname{Re}z-(\operatorname{Im}z)i.$$

Meaning: The complex conjugate reverses the sign of the imaginary part.

3) Absolute Value

Absolute Value

For $z\in\mathbb{C}$,

$$|z| = \sqrt{(\operatorname{Re}z)^2+(\operatorname{Im}z)^2}.$$

Meaning: $|z|$ is the distance from $z$ to the origin in the complex plane.

4) Properties of Complex Numbers

Conjugate Product

$$z\overline{z}=|z|^2.$$

Meaning: Multiplying a complex number by its conjugate gives the square of its absolute value.

Conjugate of a Sum

$$\overline{w+z} = \overline{w}+\overline{z}.$$

Meaning: Complex conjugation preserves addition.

Conjugate of a Product

$$\overline{wz} = \overline{w}\,\overline{z}.$$

Meaning: Complex conjugation preserves multiplication.

Double Conjugate

$$\overline{\overline{z}}=z.$$

Meaning: Taking the conjugate twice returns the original number.

Absolute Value of a Product

$$|wz| = |w||z|.$$

Meaning: Absolute values multiply when complex numbers multiply.

Triangle Inequality

$$|w+z| \le |w|+|z|.$$

Meaning: The magnitude of a sum cannot exceed the sum of the magnitudes.

4B. Zeros of Polynomials

1) Polynomial

Polynomial

A polynomial of degree $m$ over $\mathbb{F}$ has the form

$$p(z) = a_0+a_1z+\cdots+a_mz^m,$$

where

$$a_m\neq0.$$

Meaning: The degree is determined by the highest power having a nonzero coefficient.

2) Zero of a Polynomial

Zero

A number $\lambda\in\mathbb{F}$ is a zero, or root, of $p$ if

$$p(\lambda)=0.$$

Meaning: A zero is an input that makes the polynomial equal to zero.

3) Factor Corresponding to a Zero

Factor Theorem

If $p$ has degree $m$, then

$$p(\lambda)=0$$

if and only if there exists a polynomial $q$ of degree $m-1$ such that

$$p(z) = (z-\lambda)q(z).$$

Meaning: $\lambda$ is a zero exactly when $z-\lambda$ is a factor of the polynomial.

4) Number of Zeros

Degree Bound on Zeros

A nonzero polynomial of degree $m$ has at most $m$ distinct zeros.

Meaning: A polynomial cannot have more distinct roots than its degree.

5) Degree of the Zero Polynomial

Zero Polynomial

The zero polynomial is defined to have degree

$$-\infty.$$

Meaning: This convention makes degree formulas work without special exceptions.

4C. Division Algorithm for Polynomials

1) Division Algorithm

Division Algorithm

Suppose

$$p,s\in\mathcal{P}(\mathbb{F})$$

and

$$s\neq0.$$

Then there exist unique polynomials $q$ and $r$ such that

$$p=sq+r$$

and

$$\deg r<\deg s.$$

Meaning: Dividing one polynomial by another gives a unique quotient and remainder of smaller degree.

4D. Factorization over $\mathbb{C}$

1) Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

Every nonconstant polynomial with complex coefficients has at least one zero in $\mathbb{C}$.

Meaning: Complex numbers contain a root of every nonconstant complex polynomial.

2) Complete Factorization over $\mathbb{C}$

Complex Factorization

If

$$p\in\mathcal{P}(\mathbb{C})$$

is nonconstant with degree $m$, then

$$p(z) = c(z-\lambda_1)\cdots(z-\lambda_m),$$

where

$$c,\lambda_1,\ldots,\lambda_m\in\mathbb{C}.$$

The factorization is unique except for the order of the factors.

Meaning: Every complex polynomial splits completely into degree-one factors.

3) Zeros with Multiplicity

Zeros in the Factorization

In

$$p(z) = c(z-\lambda_1)\cdots(z-\lambda_m),$$

the numbers

$$\lambda_1,\ldots,\lambda_m$$

are the zeros of $p$, counted according to how many times their factors occur.

Meaning: A degree-$m$ complex polynomial has $m$ roots when repeated roots are counted.

4E. Factorization over $\mathbb{R}$

1) Conjugate Pairs of Zeros

Conjugate-Pair Property

If a polynomial has real coefficients and

$$\lambda\in\mathbb{C}$$

is a zero, then

$$\overline{\lambda}$$

is also a zero.

Meaning: Nonreal roots of real polynomials always occur in conjugate pairs.

2) Real Factor from a Conjugate Pair

Conjugate-Pair Factor

For a nonreal $\lambda\in\mathbb{C}$,

$$(x-\lambda)(x-\overline{\lambda}) = x^2 - 2(\operatorname{Re}\lambda)x + |\lambda|^2.$$

Meaning: A pair of complex-conjugate linear factors combines into a quadratic with real coefficients.

3) Factorization of a Real Quadratic

Real Quadratic Factorization

For

$$x^2+bx+c,$$

there exists a factorization

$$x^2+bx+c = (x-\lambda_1)(x-\lambda_2)$$

with $\lambda_1,\lambda_2\in\mathbb{R}$ if and only if

$$b^2\ge4c.$$

Meaning: A real quadratic factors into real linear factors exactly when its discriminant is nonnegative.

4) Complete Factorization over $\mathbb{R}$

Real Polynomial Factorization

Every nonconstant polynomial

$$p\in\mathcal{P}(\mathbb{R})$$

has a unique factorization, except for the order of factors, of the form

$$p(x) = c (x-\lambda_1)\cdots(x-\lambda_m) (x^2+b_1x+c_1) \cdots (x^2+b_Mx+c_M),$$

where all coefficients are real and

$$b_k^2<4c_k$$

for each quadratic factor.

Meaning: Every real polynomial factors into real linear factors and irreducible real quadratic factors.

5) Irreducible Quadratic Factor

Irreducible Quadratic

A quadratic factor

$$x^2+bx+c$$

with

$$b^2<4c$$

has no real zeros.

Meaning: Such a quadratic cannot be factored further into real linear factors.