Math 361, Spring 2022, Assignment 9

Read:

1. Section 23, first two pages (on 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. Constant polynomial.
3. Divisibility relation on polynomials.
4. $f\,\%\,g$.

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

1. Degree bounds on sum and product (general form).
2. Formula for $\mathrm{deg}(fg)$ when $R$ is an integral domain.
3. Theorem concerning zero-divisors in $D[x]$ when $D$ is an integral domain (i.e. "If $D$ is an integral domain then so is...")
4. Theorem on polynomial long division.
5. Divisibility test for polynomials with coefficients in a field.

Solve the following problems:

1. Section 23, problems 1, 2, 3, and 4.
2. Working in $\mathbb{Q}[x]$, find the remainder when $f(x)=x^2+x-3$ is divided by $x-5$. Then compute $f(5)$.
3. Working in $\mathbb{Z}_7[x]$, find the remainder when $f(x)=x^3+4x+1$ is divided by $x-2$. Then compute $f(2)$.
4. Using the theorem on polynomial long division, prove the conjecture suggested by the last two exercises.
5. 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 $x-a$).
6. A root of a polynomial $f\in F[x]$ is an element $a\in F$ such that $f(a)=0$. Prove that a polynomial of degree $n$ has at most $n$ roots. (Hint: begin by assuming that the roots of $f$ are $a_1,\dots,a_r$ and then prove that $\mathrm{deg}(f)\geq r$.)

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