TAOCP 4.6 Exercise 2
(a) Let $u(x)$ and $v(x)$ be monic polynomials, with leading coefficients $\ell(u) = \ell(v) = 1$.
Section 4.6: Polynomial Arithmetic
Exercise 2. [17] True or false: (a) The product of monic polynomials is monic. (b) The product of polynomials of degrees $m$ and $n$ has degree $m+n$. (c) The sum of polynomials of degrees $m$ and $n$ has degree $\max(m, n)$.
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Solution
(a) Let $u(x)$ and $v(x)$ be monic polynomials, with leading coefficients $\ell(u) = \ell(v) = 1$. Consider their product
$w(x) = u(x)v(x) = \sum_{k=0}^{\deg(u)+\deg(v)} w_k x^k,$
where $w_{\deg(u)+\deg(v)} = u_{\deg(u)} v_{\deg(v)}$ by equation (4). Since $u_{\deg(u)} = v_{\deg(v)} = 1$, it follows that $w_{\deg(u)+\deg(v)} = 1$, so $w(x)$ is monic. Therefore the statement is true.
(b) Let $u(x)$ have degree $m$ and $v(x)$ have degree $n$. By equation (4) the highest-degree term of $w(x) = u(x)v(x)$ is $w_{m+n} = u_m v_n$. If $u_m \ne 0$ and $v_n \ne 0$, then $w_{m+n} \ne 0$, so $\deg(u v) = m + n$. Hence the statement is true.
(c) Let $u(x)$ have degree $m$ and $v(x)$ have degree $n$. Consider $w(x) = u(x) + v(x)$. The coefficient of $x^k$ in $w(x)$ is $w_k = u_k + v_k$. If $m \ne n$, then $\ell(u)$ or $\ell(v)$ is the coefficient of the highest-degree term, so $\deg(u+v) = \max(m, n)$. If $m = n$, then the leading coefficients $\ell(u) + \ell(v)$ might be zero, in which case $\deg(u+v) < m = n$. Therefore the statement is false in general.
This completes the proof.
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