Question

show that the products of two monic polynomial is also monic polynomial​

Answer 1

Monic Polynomial

Definition. The polynomials in only one variable with leading coefficient $1$, are called monic polynomials.

Solution:

Let two monic polynomials be

$\quad f(x)=x^{m}+c_{1}x^{m-1}+c_{2}x^{m-2}+...+c_{m-1}$

$\quad g(x)=x^{n}+d_{1}x^{n-1}+d_{2}x^{m-2}+...+d_{n-1}$

where $c_{i},\:d_{j}\in\mathbb{R}\:\text{or}\:\in\mathbb{C}$

If the products of the polynomials $f(x)$ and $g(x)$ be defined by $h(x)$, we can write

$\quad h(x)=f(x)\times g(x)$

$\Rightarrow h(x)=(x^{m}+c_{1}x^{m-1}+...+c_{m-1})(x^{n}+c_{1}x^{n-1}+...+d_{n-1}$

$\Rightarrow h(x)=x^{m+n}+c_{1}x^{m+n-1}...+c_{m-1}d_{n-1}$

Here $h(x)$ has only one variable and the leading coefficient is $1$, and thus $h(x)$ is also monic.

This completes the proof.

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Answer 2

show that the product of two monic polynomial is a polynomial