### Matrix calculus: Minimal polynomial

### Division with remainder for polynomials

We need division with remainder for polynomials in order to discuss the notion of *minimal polynomial for matrices and linear maps*.

Division with remainder for polynomials is slightly more complicated than addition, subtraction and multiplication, especially since not every division yields a polynomial. This process is comparable to division with remainder for integers. The integer case involves a comparison of integers by their size (absolute value). For polynomials, the degree is the appropriate measure.

We recall that the *degree* of a polynomial in #x# is the highest exponent of #x# occurring in the **expanded form** (as a sum of products of a coefficient with a power of #x# having a non-negative exponent) of that polynomial. We denote the degree of the polynomial #f(x)# by #\text{deg}(f(x))#.

If \(f(x)= 0\), the zero polynomial, it is practical to define the degree of #f(x)# to be \(-1 \). If #f# is a nonzero polynomial of degree #n#, the coefficient of #x^n# in the expanded form of #f(x)# is called the **leading coefficient** of #f(x)#. A polynomial is said to be **monic** if its leading coefficient is equal to \(1\).

A polynomial of degree \(1\) is called **linear**. A polynomial of degree \(2\) is called **quadratic**.

Division with remainder starts with the definition of divisors:

Divisors of polynomials A polynomial #g(x)# is called a **divisor** of a polynomial #f(x)# if there is a polynomial #q(x)# such that #f(x) = q(x)\cdot g(x)#. In that case, #g(x)# **divides** #f(x)#. If #g(x)\ne0#, the polynomial #q(x)# is unique. It is called the **quotient** of #f(x)# and #g(x)#.

If the degree of #g(x)# is smaller than the degree of #f(x)#, then #g(x)# is called a **proper** divisor.

In order to find divisors, we use the equivalent of long division for the integers.

Division with remainder for polynomials Let #f(x)# be a polynomial of degree #m\ge1# and #g(x)# a polynomial of degree #n\ge0#. We can test as follows whether #g(x)# is a divisor of #f(x)#. Denote by #b# the leading coefficient of #g(x)#.

There exist unique polynomials #q(x)# and #r(x)# such that #f(x) = q(x)\cdot g(x) + r(x)# and the degree of #r(x)# is smaller than #n#.

The polynomial #q(x)# can be found by starting with #q(x) =0# and #r(x) = f(x)#, and subsequently

- as long as the degree #k# of #r(x)# is at least #n#, subtracting from #r(x)# the multiple #\dfrac{a}{b}x^{k-n}\cdot g(x)# of #g(x)#, where #a# is the leading coefficient of #r(x)#, and adding #\dfrac{a}{b}x^{k-n}# to #q(x)#.

The polynomial #g(x)# is a divisor of #f(x)# if and only if #r(x) = 0#.

*dividing with remainder*#f(x) = 2 x^3+3 x^2+5 x+3# by #g(x) = x^2+2 x+3#.

Do this by entering pairs #\rv{q(x),r(x)}# such that #f(x) = q(x)\cdot g(x) + r(x)# and the degree of #r(x)# keeps getting smaller.

To see this, we follow the theory and start with #q(x) =0# and #r(x) = f(x) = 2 x^3+3 x^2+5 x+3#. The degree of #r(x)# is bigger than #2#, the degree of #g(x)=x^2+2 x+3#. Therefore, we subtract #2 x \cdot g(x)# from #r(x)# and add #2 x# to #q(x)#. This gives \[\rv{q(x),r(x)} = \rv{2 x,-x^2-x+3}\] The degree of #r(x)# is not yet smaller than #2#, so we repeat this process: we subtract #-1\cdot g(x)# from #r(x)# and add #-1# to #q(x)#. This gives \[\rv{q(x),r(x)} = \rv{2 x-1,x+6}\]Now the degree of #r(x)# is smaller than #2#. The conclusion is that we have found the required #q(x)# and #r(x)#.

Since #r(x)\ne0#, the polynomial #g(x)# is not a divisor of #f(x)#.

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