Math

Advanced MathQ&A Libraryb Prove that T is not injective ifff and g have a common nontrivial factorProblem 7.8. Given a field K (say K = R or K = C), given any two polynomialsp(X), q(X) E K[X], we says that q(X) divides p(X) (and that p(X) is a multiple of q(X))iff there is somec Prove that f and g have a nontrivial common factor iff Rf,g) = 0, where R(f, g) isthe determinant given bypolynomial s(X) e K[X] such that0mP(X) q(X)s(X)000арamIn this case we say that q(X) is a factor of p(X), and if q(X) has degree at least one, wesay that q(X) is a nontrivial factor of p(X-.Rf.g)Let f(X) and g(X) be two polynomials in K[X] with-0apamb,00f(X) aoX a1X1...+a,.. .0bo...-of degree m1 andbo bb100g(X) boX bX-1+ ... + b.....The above determinant is called the resultant off and gof degree n 1 (with ao, bo 0).You will need the following result which you need not prove:Two polynomials f(X) and g(X) with deg(f) = m 2 1 and deg(g) = n 2 1 have somecommon nontrivial factor iff there erist two nonzero polynomials p(X) and q(X) such thatNote that the matrix of the resultant is an(n + m) x (n + m) matrix, with the first row(involving the a,s) occurring n times, each time shifted over to the right by one column, andthe (n1th row (involving the b,s) occurring m times, each time shifted over to the rightby one columnHint. Express the matrix of T over some suitable basis.fp g4(2) Compute the resultant in the following three cases(a) m n 1, and write f(X) aX +b and g(X) cX +dwith deg(p)n1 and deg(g) m1(b) m 1 and n 2 2 arbitrary(1) Let Pm denote the vector space of all polynomials in K[X] of degree at most m - 1,and let T: P x PmPmtn be the map given by>(c) f(X) aX2+bX +c and g(X) 2aX +b.Т(р,9) — fp+ g9, р€Pа, q€ Рm,(3) Compute the resultant of f(X)= X3 +pX q and g(X)= 3X2 p, andfixed polynomials of degree mwhere f and q are somef(X) aoX aj X +a2g(X) bXb X + b21 and n 2 1.a) Prove that the map T is linearIn the second case, you should get4R(f.9)= (2agb2 -a,b+2agbn) - (4agaz - ai)(4bpb2 - t).Question

Asked Sep 28, 2019

2 views

Please, help me with a step by step solutions to problem (Q2 a,b,c) and I will be very grateful to you. I don't have a strong background in algebra. I hope you will help.

Step 1

To compute the resolvents for the given pairs of polynomials

Step 2

We use the fact that res(f,g) is 0 iff f(x) and g(x) have a common factor.

Step 3

Tagged in

Find answers to questions asked by student like you

Show more Q&A

Q: Please help me solve problem 20. Thanks!

A: To prove that the vector v= ||y||x+||x||y always bisects the angle between the vectors x and y; that...

Q: Mail Tucker, Ge x Limiting Reactan lc Wait for Next Q X b Login | bartleby Bb MyLab Math - R X WA4.d...

A: Let the f:R→R is defined by f(x)=rx+d where r and d are real constants.Then we have to show that f i...

Q: 32

A: To ddscribe the lattice structure of the set of all subgroups of the group Z12

Q: I need help for the proof of question 2. Thank you very much!

A: Given:

Q: Suppose {Iα}α∈A is a collection of open intervals that are mutually disjoint: If α1 ≠ α2, then Iα1 ∩...

A: Consider the given information:The collection {{Iα}, where α belongs to A} of open intervals that ar...

Q: Show that the set {5, 15, 25, 35} is a group under multiplication modulo 40. What is the identity el...

A: Calculation:Obtain the Cayley table for the set S = {5,15,25,35} under the multiplication modulo 40 ...

Q: Find an equation of the line that passes through the point (5, 3) and is perpendicular to the line 2...

A: We need to find an equation of the line that passes through the point (5, 3) and is perpendicular to...

Q: How many integers greater than 5400 and less than 10000 have both of the following properties: - dig...

A: To count the number of positive integers satisfying all the required properties

Q: In the following problems, decide if the groups G and G are isomorphic. If they are not, give proper...

A: (a) We are given that G = GL(2, R), the group of 2 × 2 non-singular matrices under multiplication;...