Question

In each part, give an example of polynomials $f\left(x\right),g\left(x\right)\in \mathrm{\mathbb{Q}}\left[x\right]$ that satisfy the given condition:

1. The degree of $f\left(x\right)+g\left(x\right)$ is less than the maximum of $degf\left(x\right)anddegg\left(x\right)$.
2. $Deg\left[f\left(x\right)+g\left(x\right)\right]=max\left\{degf\left(x\right),degg\left(x\right)\right\}$

Short answer

The required polynomials are:

1. $$\begin{gathered} f\left(x\right)=x^2+x \\ g\left(x\right)=-x^2 \end{gathered}$$

2. $$\begin{gathered} f\left(x\right)=x^2 \\ g\left(x\right)=x \end{gathered}$$

Step-by-step solution

Polynomial Arithmetic:

If any given function $R\left[x\right]$ is a ring, then the commutative, associative, and distributive laws hold such that the function$f\left(x\right)+g\left(x\right)$ exists.

Polynomials:

(a)

Let us assume the polynomials such that:

$$\begin{gathered} \left\{f\left(x\right)=x^2+x \\ g\left(x\right)=-x^2\right\}\in \mathrm{\mathbb{Q}}\left[x\right] \end{gathered}$$

Now, for the degrees of polynomials as:

$deg\left[f\left(x\right)+g\left(x\right)\right]=deg\left[x^2+x-x^2\right]=deg\left[x\right]=1$

And, the maximum degree:

$max\left\{deg\left[f\left(x\right)\right],deg\left[g\left(x\right)\right]\right\}=max\left\{\left[2\right],\left[2\right]\right\}=2$

Hence, for these polynomials, we have:

$deg\left[f\left(x\right)+g\left(x\right)\right]<max\left\{deg\left[f\left(x\right)\right],deg\left[g\left(x\right)\right]\right\}$

Polynomials: 

(b)

Let us assume the polynomials such that:

$$\begin{gathered} \left\{f\left(x\right)=x^2 \\ g\left(x\right)=x\right\}\in \mathrm{\mathbb{Q}}\left[x\right] \end{gathered}$$

Now, for the degrees of polynomials as:

$deg\left[f\left(x\right)+g\left(x\right)\right]=deg\left[x^2+x\right]=2$

And, the maximum degree:

$maxdeg\left\{\left[f\left(x\right)\right],\left[g\left(x\right)\right]\right\}=max\left\{\left[2\right],\left[1\right]\right\}=2$

Hence, for these polynomials, we have:

$deg\left[f\left(x\right)+g\left(x\right)\right]=max\left\{deg\left[f\left(x\right)\right],deg\left[g\left(x\right)\right]\right\}$