Real Polynomials

Definition: Real Polynomial

A real polynomial is a polynomial over the field of the real numbers.

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content [!THEOREM]- The Binomial Theorem

The expansion of the expression

$$(x+y{)}^n,$$

where $n\in \mathbb{N}$ is given by the real polynomials

$$(x+y{)}^n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^{n-k}{y}^k=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^k{y}^{n-k}$$

TIP

The expansion has $n+1$ terms. The $k$-th term ($k\in \{1,\dots ,n+1\}$) is $C_n^{k-1}{x}^{n-(k-1)}{y}^{k-1}$.

PROOF

TODO

Polynomial Division

Theorem: Polynomial Division

Let $A(x)$ and $B(x)$ be two Real Polynomials such that the degree of $A$ is greater than or equal to the degree of $B$.

If $B(x)$ is nonzero, then there exist unique polynomials $Q(x)$ and $R(x)$ such that

$$A(x)=Q(x)B(x)+R(x),$$

where

• $\mathrm{deg}(Q)=\mathrm{deg}(A)-\mathrm{deg}(B)$
• $\mathrm{deg}(R)<\mathrm{deg}(B)$ or $R(x)$ is the zero polynomial.

We call $A$ the dividend, $B$ the divisor, $Q$ the quotient and $R$ the remainder.

PROOF

TODO

Definition: Divisibility

If $R(x)=0$, then we say that $A$ is divisible by $B$.

Theorem: Factorization Theorem

Every real polynomial $A(x)={\sum }_{k=0}^n{a}_k{x}^k$ can be factorized into a product of real polynomials of degree $\le 2$. More specifically,

$$A(x)=a_n(x-r_1{)}^{p_1}\cdots (x-r_l{)}^{p_l}(x^2+c_{1}x+d_1{)}^{q_1}\cdots (x^2+c_{s}x+d_s{)}^{q_s},$$

where:

• $r_i$ are the distinct real roots of $A(x)=0$;

• $p_i$ is the multiplicity of $r_i$;

• $c_j,d_j\in \mathbb{R}$;

• $q_j\in {\mathbb{N}}_0$.

PROOF

TODO

Polynomial Remainder Theorem (Little Bézout's Theorem)

The remainder $R$ upon the division of a real polynomial $A(x)$ with a real polynomial $B(x)=x-p$ is the value of $A$ at $p$.

$$R=A(p)$$

PROOF

The polynomial division theorem tells us that

$$A(x)=(x-p)Q(x)+R(x)$$

Since the degree of $B(x)$ is $1$, the degree of $R$ must either be $1$, i.e. $R$ is a constant and contains no variables or $R$ must be zero. We can therefore denote $R(x)$ as just $R$. For $x=p$ we obtain

$$A(p)=(p-p)Q(p)+R=0+R=R$$

Algorithm: Horner's Method

Horner’s method is a way to easily divide a polynomial $A(x)$ by a polynomial $B(x)$ of degree one.

1. Write $B(x)$ as $B(x)=x-p$. Since $\mathrm{deg}(B)=1$, the remainder $R$ is just a real number because $\mathrm{deg}(R)<\mathrm{deg}(B)⟹\mathrm{deg}(R)=0$. This means that we have

$$A(x)=(x-p)Q(x)+R$$

2. To determine $Q(x)$ and $R$, construct the following table:

	$a_n$	$a_{n-1}$	$\cdots$	$a_1$	$a_0$
$p$	$q_{n-1}=a_n$	$q_{n-2}=p{q}_{n-1}+a_{n-1}$	$\cdots$	$q_0=p{q}_1+a_1$	$R=p{q}_0+a_0$

• We calculate the table from left to right.

• The first coefficient $q_{n-1}$ of $Q(x)$ is equal to the first coefficient of $A(x)$, i.e. $q_{n-1}=a_n$.

• For all other coefficients of $Q(x)$ we have $q_i=p{q}_{i+1}+a_{i-1}$.

• At the end of the calculation, the right-most cell will contain $R$.

EXAMPLE

Let $A(x)=2x^5+3x^3-11x^2+6$ and $B(x)=x-3$. Create the table.

	$2$	$0$	$3$	$-11$	$0$	$6$
$p=3$

Perform the algorithm step by step.

	$2$	$0$	$3$	$-11$	$0$	$6$
$p=3$	2

	$2$	$0$	$3$	$-11$	$0$	$6$
$p=3$	2	6

	$2$	$0$	$3$	$-11$	$0$	$6$

Algorithm: Change of Variables

We are given a real polynomial $A(x)$ and want to transform it into another real polynomial $B(y)$ where $y=x-p$ for some $p\in \mathbb{R}$.

1. Divide $A$ by $(x-p)$.

$$A(x)=(x-p)Q_1(x)+R_0$$

2. Divide $Q_1$ by $(x-p)$.

$$Q_1(x)=(x-p)Q_2(x)+R_1$$

3. Repeat step 2, dividing $Q_i$ by $(x-p)$ in order to obtain $Q_{i+1}$ and $R_1$ until $Q_{i+1}$ is just a number.
4. Finally,

$$B(y)=R_0+R_{1}y+R_2{y}^2+R_3{y}^3+\cdots$$

TIP

You can use Horner’s method to speed up the process.

EXAMPLE

Let $A(x)=x^4+8x^3+24x^2+50x+90$ and let $y=x+2$. We can write $y$ as $y=x-p$, where $p=-2$.

Apply Horner’s method:

	$1$	$8$	$24$	$50$	$90$
$-2$	$1$	$6$	$12$	$26$	$38=R_0$
$-2$	$1$	$4$	$4$	$18=R_1$
$-2$	$1$	$2$	$0=R_2$
$-2$	$1$	$0=R_3$
$-2$	$1=R_4$

We have

$$\begin{array}{rl}B(y)& =R_0+R_{1}y+R_2{y}^2+R_3{y}^3+R_4{y}^4\\ & =38+18y+y^4\\ & =y^4+18y+38\\ & =(x+2{)}^4+18(x+2)+38\end{array}$$