Answered: 17. Find the solution to the… | bartleby

Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.2: Determinants

Problem 16AEXP

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Question

17. Find the solution to the interpolation problem of finding a polynomial q (x) with
deg(q) < 2 and such that
q(xo) =
= yo,
q(x1) = yı,
q'(x1) = yi
with xo x1.

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Find a polynomial of degree n=2 that has the given zero(s) x = −3
I am trying to find out if the polynomial 12x^4+8x^3+9x^2+4x+4 is reducible or irreducible over Q. We can't apply Eisenstein's criterion here. But we can try this:12x^4+8x^3+9x^2+4x+4 = x^4+(2/3)x^3+(3/4)x^2+(1/3)x+(1/3) x^4+(2/3)x^3+(3/4)x^2+(1/3)x+(1/3)=(x^2+ax+b)(x^2+cx+d) = x^4+(a+c)x^3+(b+d+ac)x^2+(ad+dc)x+bd We get the equation system: a+c=(2/3)b+d+ac=(3/4)ab+bc=(1/3)bd=(1/3) I get the following result:b=(1/2)d=(2/3)c=(2+sqrt(19))/6 and (2-sqrt(19))/6a=(2/3)-((2+sqrt(19))/6) and (2/3)-((2-sqrt(19))/6) The expression in a contains an irrational number, sqrt(19), this means that the polynomial is irreversible over Q. Is this correct?
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