Math 361, Spring 2021, Assignment 9

Read:

1. Section 23, first two pages (concerning the division algorithm).

Carefully define the following terms, and give one example and one non-example of each:

1. Degree (of a polynomial; please be sure to include the case of the zero polynomial).
2. Leading coefficient (of a non-zero polynomial).
3. Constant polynomial.
4. Canonical injection (of $R$ into $R[x]$).

Carefully state the following theorems (you do not need to prove them):

1. Degree bounds on sum and product (general form).
2. Formula for $\mathrm{deg}(f+g)$ when $\mathrm{deg}(f)\neq\mathrm{deg}(g)$.
3. Formula for $\mathrm{deg}(fg)$ when $R$ is an integral domain.
4. Theorem concerning zero-divisors in $D[x]$ when $D$ is an integral domain (i.e. "If $D$ is an integral domain then so is...")
5. Theorem concerning zero-divisors in $D[x_1,\dots,x_n]$ when $D$ is an integral domain.
6. Universal mapping property of $R[x]$.
7. Theorem on polynomial long division.

Carefully describe the following algorithms:

1. Polynomial long division algorithm.

Solve the following problems:

1. Section 22, problems 7, 9, 11, 20, and 25.
2. Section 23, problems 1 and 3.
3. Working in $\mathbb{Q}[x]$, find the remainder when $f(x)=x^2+x-3$ is divided by $x-5$. Then compute $f(5)$.
4. Working in $\mathbb{Z}_7[x]$, find the remainder when $f(x)=x^3+4x+1$ is divided by $x-2$. Then compute $f(2)$.
5. Using the theorem on polynomial long division, prove the conjecture suggested by the last two exercises.
6. Prove the Factor Theorem: if $F$ is any field, and $f\in F[x]$ is any polynomial with coefficients in $F$, then $f(a)=0$ if and only if $x-a$ is a factor of $f$ (i.e. $f$ is a multiple of $x-a$).

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