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When we have a polynomial with integer coefficients,the following theorem provides us a good

By December 6, 2021No Comments

When we have a polynomial with integer coefficients,the following theorem provides us a goodway to find possible rational roots for polynomials:Rational Roots Theorem (RRT)For all polynomials f(x)=anxn+an-1xn-1+..+a1x+ao with integer coefficients and n≥1,if is a rational root of f(z)with ged(p,g)=1,then p|ao and q l an.One can find the proof of the theorem in Section 11.5.To see a use of the rational root theorem,we consider f(x)=2×3+5×2-1.Letp/q∈Q with ged(p,q)=1 be a root of f(x).By theRational Root Theorem,p|(-1)andq2.It follows that p∈{±l}andq∈{+1,2}.Hencethe possible rational roots of f(x)are±land±.Note that工-11-1/21/2f(x)2601/2Thus r=is the only rational root.QuestionFactor f(z)=2×4+x3-5×2-2x+2 into a product of irreducible polynomials in Qz].

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