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Polynomials: exercises

Polynomials: exercises — from Judson, Abstract Algebra: Theory and Applications.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. List all of the polynomials of degree \(3\) or less in \({\mathbb Z}_2[x]\).

  2. Compute each of the following.

    1. \((5x^2 + 3x - 4) + (4x^2 - x + 9)\) in \({\mathbb Z}_{12}[x]\)

    2. \((5x^2 + 3x - 4) (4x^2 - x + 9)\) in \({\mathbb Z}_{12}[x]\)

    3. \((7x^3 + 3x^2 - x) + (6x^2 - 8x + 4)\) in \({\mathbb Z}_9[x]\)

    4. \((3x^2 + 2x - 4) + (4x^2 + 2)\) in \({\mathbb Z}_5[x]\)

    5. \((3x^2 + 2x - 4) (4x^2 + 2)\) in \({\mathbb Z}_5[x]\)

    6. \((5x^2 + 3x - 2)^2\) in \({\mathbb Z}_{12}[x]\)

    Jiżvelaw it-tweġiba

    Hint:

    (a) \(9x^2 + 2x + 5\); (b) \(8x^4 + 7x^3 + 2x^2 + 7x\).

  3. Use the division algorithm to find \(q(x)\) and \(r(x)\) such that \(a(x) = q(x) b(x) + r(x)\) with \(\deg r(x) \lt \deg b(x)\) for each of the following pairs of polynomials.

    1. \(a(x) = 5 x^3 + 6x^2 - 3 x + 4\) and \(b(x) = x - 2\) in \({\mathbb Z}_7[x]\)

    2. \(a(x) = 6 x^4 - 2 x^3 + x^2 - 3 x + 1\) and \(b(x) = x^2 + x - 2\) in \({\mathbb Z}_7[x]\)

    3. \(a(x) = 4 x^5 - x^3 + x^2 + 4\) and \(b(x) = x^3 - 2\) in \({\mathbb Z}_5[x]\)

    4. \(a(x) = x^5 + x^3 -x^2 - x\) and \(b(x) = x^3 + x\) in \({\mathbb Z}_2[x]\)

    Jiżvelaw it-tweġiba

    Hint:

    (a) \(5 x^3 + 6 x^2 - 3 x + 4 = (5 x^2 + 2x + 1)(x -2) + 6\); (c) \(4x^5 - x^3 + x^2 + 4 = (4x^2 + 4)(x^3 + 3) + 4x^2 + 2\).

  4. Find the greatest common divisor of each of the following pairs \(p(x)\) and \(q(x)\) of polynomials. If \(d(x) = \gcd( p(x), q(x) )\), find two polynomials \(a(x)\) and \(b(x)\) such that \(a(x) p(x) + b(x) q(x) = d(x)\).

    1. \(p(x) = x^3 - 6x^2 + 14x - 15\) and \(q(x) = x^3 - 8x^2 + 21x - 18\), where \(p(x), q(x) \in {\mathbb Q}[x]\)

    2. \(p(x) = x^3 + x^2 - x + 1\) and \(q(x) = x^3 + x - 1\), where \(p(x), q(x) \in {\mathbb Z}_2[x]\)

    3. \(p(x) = x^3 + x^2 - 4x + 4\) and \(q(x) = x^3 + 3 x -2\), where \(p(x), q(x) \in {\mathbb Z}_5[x]\)

    4. \(p(x) = x^3 - 2 x + 4\) and \(q(x) = 4 x^3 + x + 3\), where \(p(x), q(x) \in {\mathbb Q}[x]\)

  5. Find all of the zeros for each of the following polynomials.

    1. \(5x^3 + 4x^2 - x + 9\) in \({\mathbb Z}_{12}[x]\)

    2. \(3x^3 - 4x^2 - x + 4\) in \({\mathbb Z}_{5}[x]\)

    3. \(5x^4 + 2x^2 - 3\) in \({\mathbb Z}_{7}[x]\)

    4. \(x^3 + x + 1\) in \({\mathbb Z}_2[x]\)

    Jiżvelaw it-tweġiba

    Hint:

    (a) No zeros in \({\mathbb Z}_{12}\); (c) \(3\), \(4\).

  6. Find all of the units in \({\mathbb Z}[x]\).

  7. Find a unit \(p(x)\) in \({\mathbb Z}_4[x]\) such that \(\deg p(x) \gt 1\).

    Jiżvelaw it-tweġiba

    Hint:

    Look at \((2x + 1)\).

  8. Which of the following polynomials are irreducible over \({\mathbb Q}[x]\)?

    1. \(x^4 - 2x^3 + 2x^2 + x + 4\)

    2. \(x^4 - 5x^3 + 3x - 2\)

    3. \(3x^5 - 4x^3 - 6x^2 + 6\)

    4. \(5x^5 - 6x^4 - 3x^2 + 9 x - 15\)

    Jiżvelaw it-tweġiba

    Hint:

    (a) Reducible; (c) irreducible.

  9. Find all of the irreducible polynomials of degrees \(2\) and \(3\) in \({\mathbb Z}_2[x]\).

  10. Give two different factorizations of \(x^2 + x + 8\) in \({\mathbb Z}_{10}[x]\).

    Jiżvelaw it-tweġiba

    Hint:

    One factorization is \(x^2 + x + 8 = (x + 2)(x + 9)\).

  11. Prove or disprove: There exists a polynomial \(p(x)\) in \({\mathbb Z}_6[x]\) of degree \(n\) with more than \(n\) distinct zeros.

  12. If \(F\) is a field, show that \(F[x_1, \ldots, x_n]\) is an integral domain.

  13. Show that the division algorithm does not hold for \({\mathbb Z}[x]\). Why does it fail?

    Jiżvelaw it-tweġiba

    Hint:

    The integers \(\mathbb Z\) do not form a field.

  14. Prove or disprove: \(x^p + a\) is irreducible for any \(a \in {\mathbb Z}_p\), where \(p\) is prime.

    Jiżvelaw it-tweġiba

    Hint:

    False.

  15. Let \(f(x)\) be irreducible in \(F[x]\), where \(F\) is a field. If \(f(x) \mid p(x)q(x)\), prove that either \(f(x) \mid p(x)\) or \(f(x) \mid q(x)\).

  16. Suppose that \(R\) and \(S\) are isomorphic rings. Prove that \(R[x] \cong S[x]\).

    Jiżvelaw it-tweġiba

    Hint:

    Let \(\phi : R \rightarrow S\) be an isomorphism. Define \(\overline{\phi} : R[x] \rightarrow S[x]\) by \(\overline{\phi}(a_0 + a_1 x + \cdots + a_n x^n) = \phi(a_0) + \phi(a_1) x + \cdots + \phi(a_n) x^n\).

  17. Let \(F\) be a field and \(a \in F\). If \(p(x) \in F[x]\), show that \(p(a)\) is the remainder obtained when \(p(x)\) is divided by \(x - a\).

  18. Let \[\begin{aligned}\end{aligned}\], where \(a_n \neq 0\). Prove that if \(p(r/s) = 0\), where \(\gcd(r, s) = 1\), then \(r \mid a_0\) and \(s \mid a_n\).

  19. Let \({\mathbb Q}^*\) be the multiplicative group of positive rational numbers. Prove that \({\mathbb Q}^*\) is isomorphic to \(( {\mathbb Z}[x], +)\).

  20. The polynomial \[\begin{aligned}\end{aligned}\] for \(p\) prime is called the cyclotomic polynomial. Show that \(\Phi_p(x)\) is irreducible over \({\mathbb Q}\) for any prime \(p\).

    Jiżvelaw it-tweġiba

    Hint:

    The polynomial \[\begin{aligned}\end{aligned}\] is called the cyclotomic polynomial. Show that \(\Phi_p(x)\) is irreducible over \({\mathbb Q}\) for any prime \(p\).

  21. If \(F\) is a field, show that there are infinitely many irreducible polynomials in \(F[x]\).

  22. Let \(R\) be a commutative ring with identity. Prove that multiplication is commutative in \(R[x]\).

  23. Let \(R\) be a commutative ring with identity. Prove that multiplication is distributive in \(R[x]\).

  24. Show that \(x^p - x\) has \(p\) distinct zeros in \({\mathbb Z}_p\), for any prime \(p\). Conclude that \[\begin{aligned}\end{aligned}\].

  25. Let \(F\) be a field and \(f(x) = a_0 + a_1 x + \cdots + a_n x^n\) be in \(F[x]\). Define \(f'(x) = a_1 + 2 a_2 x + \cdots + n a_n x^{n - 1}\) to be the derivative of \(f(x)\).

    1. Prove that \[\begin{aligned}\end{aligned}\]. Conclude that we can define a homomorphism of abelian groups \(D : F[x] \rightarrow F[x]\) by \(D(f(x)) = f'(x)\).

    2. Calculate the kernel of \(D\) if \(\chr F = 0\).

    3. Calculate the kernel of \(D\) if \(\chr F = p\).

    4. Prove that \[\begin{aligned}\end{aligned}\].

    5. Suppose that we can factor a polynomial \(f(x) \in F[x]\) into linear factors, say \[\begin{aligned}\end{aligned}\]. Prove that \(f(x)\) has no repeated factors if and only if \(f(x)\) and \(f'(x)\) are relatively prime.

  26. Let \(F\) be a field. Show that \(F[x]\) is never a field.

    Jiżvelaw it-tweġiba

    Hint:

    Find a nontrivial proper ideal in \(F[x]\).

  27. Let \(R\) be an integral domain. Prove that \(R[x_1, \ldots, x_n]\) is an integral domain.

  28. Let \(R\) be a commutative ring with identity. Show that \(R[x]\) has a subring \(R'\) isomorphic to \(R\).

  29. Let \(p(x)\) and \(q(x)\) be polynomials in \(R[x]\), where \(R\) is a commutative ring with identity. Prove that \(\deg( p(x) + q(x) ) \leq \max( \deg p(x), \deg q(x) )\).

  30. Complete the square to solve the general quadratic equation \[\begin{aligned}\end{aligned}\] to obtain \[\begin{aligned}\end{aligned}\]. The discriminant of the quadratic equation \(\Delta = b^2 - 4ac\) determines the nature of the solutions of the equation. If \(\Delta \gt 0\), the equation has two distinct real solutions. If \(\Delta = 0\), the equation has a single repeated real root. If \(\Delta \lt 0\), there are two distinct imaginary solutions.

  31. Show that any cubic equation of the form \[\begin{aligned}\end{aligned}\] can be reduced to the form \(y^3 + py + q = 0\) by making the substitution \(x = y - b/3\).

  32. Prove that the cube roots of \(1\) are given by \[\begin{aligned}\omega & = \frac{-1+ i \sqrt{3}}{2} \\ \omega^2 & = \frac{-1- i \sqrt{3}}{2} \\ \omega^3 & = 1\end{aligned}\].

  33. Make the substitution \[\begin{aligned}\end{aligned}\] for \(y\) in the equation \(y^3 + py + q = 0\) and obtain two solutions \(A\) and \(B\) for \(z^3\).

  34. Show that the product of the solutions obtained in (4) is \(-p^3/27\), deducing that \(\sqrt[3]{A B} = -p/3\).

  35. Prove that the possible solutions for \(z\) in (4) are given by \[\begin{aligned}\end{aligned}\] and use this result to show that the three possible solutions for \(y\) are \[\begin{aligned}\end{aligned}\], where \(i = 0, 1, 2\).

  36. The discriminant of the cubic equation is \[\begin{aligned}\end{aligned}\]. Show that \(y^3 + py + q=0\)

    1. has three real roots, at least two of which are equal, if \(\Delta = 0\).

    2. has one real root and two conjugate imaginary roots if \(\Delta \gt 0\).

    3. has three distinct real roots if \(\Delta \lt 0\).

  37. Solve the following cubic equations.

    1. \(x^3 - 4x^2 + 11 x + 30 = 0\)

    2. \(x^3 - 3x +5 = 0\)

    3. \(x^3 - 3x +2 = 0\)

    4. \(x^3 + x + 3 = 0\)

  38. Show that the general quartic equation \[\begin{aligned}\end{aligned}\] can be reduced to \[\begin{aligned}\end{aligned}\] by using the substitution \(x = y - a/4\).

  39. Show that \[\begin{aligned}\end{aligned}\].

  40. Show that the right-hand side of can be put in the form \((my + k)^2\) if and only if \[\begin{aligned}\end{aligned}\].

Symbols used here

a \mid b,\ \gcd(a,b)
divides, greatest common divisor
b is a multiple of a; the largest number dividing both.
x \in A,\ A \subseteq B
element of, subset
x belongs to A; every element of A is in B.
\neq
not equal
The two sides are different.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.
a \equiv b \pmod n
congruent modulo n
n divides a − b; a and b have the same remainder.
(G, \cdot),\ e,\ g^{-1}
group, identity, inverse
A set with an operation; the do-nothing element; the element that undoes g.
G \cong H,\ G / N
isomorphic, quotient group
Same structure; the group of cosets of a normal subgroup N.
\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n
integers modulo n
The remainders 0…n−1 with clock arithmetic.
\operatorname{Hom}(A, B),\ f \circ g
arrows from A to B, composition
The set of morphisms; do g then f.

Questions people ask

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group — that is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

Ipprova tiegħek stess

Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.

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