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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.
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List all of the polynomials of degree \(3\) or less in \({\mathbb Z}_2[x]\).
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Compute each of the following.
\((5x^2 + 3x - 4) + (4x^2 - x + 9)\) in \({\mathbb Z}_{12}[x]\)
\((5x^2 + 3x - 4) (4x^2 - x + 9)\) in \({\mathbb Z}_{12}[x]\)
\((7x^3 + 3x^2 - x) + (6x^2 - 8x + 4)\) in \({\mathbb Z}_9[x]\)
\((3x^2 + 2x - 4) + (4x^2 + 2)\) in \({\mathbb Z}_5[x]\)
\((3x^2 + 2x - 4) (4x^2 + 2)\) in \({\mathbb Z}_5[x]\)
\((5x^2 + 3x - 2)^2\) in \({\mathbb Z}_{12}[x]\)
Atskleisti atsakymą
Hint:
(a) \(9x^2 + 2x + 5\); (b) \(8x^4 + 7x^3 + 2x^2 + 7x\).
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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.
\(a(x) = 5 x^3 + 6x^2 - 3 x + 4\) and \(b(x) = x - 2\) in \({\mathbb Z}_7[x]\)
\(a(x) = 6 x^4 - 2 x^3 + x^2 - 3 x + 1\) and \(b(x) = x^2 + x - 2\) in \({\mathbb Z}_7[x]\)
\(a(x) = 4 x^5 - x^3 + x^2 + 4\) and \(b(x) = x^3 - 2\) in \({\mathbb Z}_5[x]\)
\(a(x) = x^5 + x^3 -x^2 - x\) and \(b(x) = x^3 + x\) in \({\mathbb Z}_2[x]\)
Atskleisti atsakymą
Hint:
(a) \(5 x^3 + 6 x^2 - 3 x + 4 = (5 x^2 + 2x + 1)(x -2) + 6\); (c) \(4x^5 - x^3 + x^2 + 4 = (4x^2 + 4)(x^3 + 3) + 4x^2 + 2\).
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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 \(a(x) p(x) + b(x) q(x) = d(x)\).
\(p(x) = x^3 - 6x^2 + 14x - 15\) and \(q(x) = x^3 - 8x^2 + 21x - 18\), where \(p(x), q(x) \in {\mathbb Q}[x]\)
\(p(x) = x^3 + x^2 - x + 1\) and \(q(x) = x^3 + x - 1\), where \(p(x), q(x) \in {\mathbb Z}_2[x]\)
\(p(x) = x^3 + x^2 - 4x + 4\) and \(q(x) = x^3 + 3 x -2\), where \(p(x), q(x) \in {\mathbb Z}_5[x]\)
\(p(x) = x^3 - 2 x + 4\) and \(q(x) = 4 x^3 + x + 3\), where \(p(x), q(x) \in {\mathbb Q}[x]\)
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Find all of the zeros for each of the following polynomials.
\(5x^3 + 4x^2 - x + 9\) in \({\mathbb Z}_{12}[x]\)
\(3x^3 - 4x^2 - x + 4\) in \({\mathbb Z}_{5}[x]\)
\(5x^4 + 2x^2 - 3\) in \({\mathbb Z}_{7}[x]\)
\(x^3 + x + 1\) in \({\mathbb Z}_2[x]\)
Atskleisti atsakymą
Hint:
(a) No zeros in \({\mathbb Z}_{12}\); (c) \(3\), \(4\).
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Find all of the units in \({\mathbb Z}[x]\).
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Find a unit \(p(x)\) in \({\mathbb Z}_4[x]\) such that \(\deg p(x) \gt 1\).
Atskleisti atsakymą
Hint:
Look at \((2x + 1)\).
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Which of the following polynomials are irreducible over \({\mathbb Q}[x]\)?
\(x^4 - 2x^3 + 2x^2 + x + 4\)
\(x^4 - 5x^3 + 3x - 2\)
\(3x^5 - 4x^3 - 6x^2 + 6\)
\(5x^5 - 6x^4 - 3x^2 + 9 x - 15\)
Atskleisti atsakymą
Hint:
(a) Reducible; (c) irreducible.
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Find all of the irreducible polynomials of degrees \(2\) and \(3\) in \({\mathbb Z}_2[x]\).
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Give two different factorizations of \(x^2 + x + 8\) in \({\mathbb Z}_{10}[x]\).
Atskleisti atsakymą
Hint:
One factorization is \(x^2 + x + 8 = (x + 2)(x + 9)\).
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Prove or disprove: There exists a polynomial \(p(x)\) in \({\mathbb Z}_6[x]\) of degree \(n\) with more than \(n\) distinct zeros.
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If \(F\) is a field, show that \(F[x_1, \ldots, x_n]\) is an integral domain.
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Show that the division algorithm does not hold for \({\mathbb Z}[x]\). Why does it fail?
Atskleisti atsakymą
Hint:
The integers \(\mathbb Z\) do not form a field.
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Prove or disprove: \(x^p + a\) is irreducible for any \(a \in {\mathbb Z}_p\), where \(p\) is prime.
Atskleisti atsakymą
Hint:
False.
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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 \(f(x) \mid q(x)\).
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Suppose that \(R\) and \(S\) are isomorphic rings. Prove that \(R[x] \cong S[x]\).
Atskleisti atsakymą
Hint:
Let \(\phi : R \rightarrow S\) be an isomorphism. Define \(\overline{\phi} : R[x] \rightarrow S[x]\) by \(\overline{\phi}(a_0 + a_1 x + \cdots + a_n x^n) = \phi(a_0) + \phi(a_1) x + \cdots + \phi(a_n) x^n\).
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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 \(x - a\).
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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 \(s \mid a_n\).
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Let \({\mathbb Q}^*\) be the multiplicative group of positive rational numbers. Prove that \({\mathbb Q}^*\) is isomorphic to \(( {\mathbb Z}[x], +)\).
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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 \(p\).
Atskleisti atsakymą
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 \(p\).
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If \(F\) is a field, show that there are infinitely many irreducible polynomials in \(F[x]\).
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Let \(R\) be a commutative ring with identity. Prove that multiplication is commutative in \(R[x]\).
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Let \(R\) be a commutative ring with identity. Prove that multiplication is distributive in \(R[x]\).
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Show that \(x^p - x\) has \(p\) distinct zeros in \({\mathbb Z}_p\), for any prime \(p\). Conclude that \[\begin{aligned}\end{aligned}\].
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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 \(f(x)\).
Prove that \[\begin{aligned}\end{aligned}\]. Conclude that we can define a homomorphism of abelian groups \(D : F[x] \rightarrow F[x]\) by \(D(f(x)) = f'(x)\).
Calculate the kernel of \(D\) if \(\chr F = 0\).
Calculate the kernel of \(D\) if \(\chr F = p\).
Prove that \[\begin{aligned}\end{aligned}\].
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.
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Let \(F\) be a field. Show that \(F[x]\) is never a field.
Atskleisti atsakymą
Hint:
Find a nontrivial proper ideal in \(F[x]\).
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Let \(R\) be an integral domain. Prove that \(R[x_1, \ldots, x_n]\) is an integral domain.
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Let \(R\) be a commutative ring with identity. Show that \(R[x]\) has a subring \(R'\) isomorphic to \(R\).
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Let \(p(x)\) and \(q(x)\) be polynomials in \(R[x]\), where \(R\) is a commutative ring with identity. Prove that \(\deg( p(x) + q(x) ) \leq \max( \deg p(x), \deg q(x) )\).
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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.
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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 \(x = y - b/3\).
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Prove that the cube roots of \(1\) are given by \[\begin{aligned}\omega & = \frac{-1+ i \sqrt{3}}{2} \\ \omega^2 & = \frac{-1- i \sqrt{3}}{2} \\ \omega^3 & = 1\end{aligned}\].
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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 \(z^3\).
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Show that the product of the solutions obtained in (4) is \(-p^3/27\), deducing that \(\sqrt[3]{A B} = -p/3\).
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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 \(i = 0, 1, 2\).
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The discriminant of the cubic equation is \[\begin{aligned}\end{aligned}\]. Show that \(y^3 + py + q=0\)
has three real roots, at least two of which are equal, if \(\Delta = 0\).
has one real root and two conjugate imaginary roots if \(\Delta \gt 0\).
has three distinct real roots if \(\Delta \lt 0\).
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Solve the following cubic equations.
\(x^3 - 4x^2 + 11 x + 30 = 0\)
\(x^3 - 3x +5 = 0\)
\(x^3 - 3x +2 = 0\)
\(x^3 + x + 3 = 0\)
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Show that the general quartic equation \[\begin{aligned}\end{aligned}\] can be reduced to \[\begin{aligned}\end{aligned}\] by using the substitution \(x = y - a/4\).
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Show that \[\begin{aligned}\end{aligned}\].
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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
b is a multiple of a; the largest number dividing both.
x belongs to A; every element of A is in B.
The two sides are different.
Naturals, integers, rationals, reals, complex numbers.
Marks the point where the statement has been established.
n divides a − b; a and b have the same remainder.
A set with an operation; the do-nothing element; the element that undoes g.
Same structure; the group of cosets of a normal subgroup N.
The remainders 0…n−1 with clock arithmetic.
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.
Pabandyk savo pačių
Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.
Daugiau informacijos Abstract Algebra
GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula