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Concept derivation guides
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Concept guide
In ℚ[x], let p(x)=4x³+3x²−x+2−4x³+x, let q have coefficient vector [−1,1,0,0], and let r(x)=x²−x²+3, z(x)=5x−5x and t(x)=5x. Find the coefficients and degrees of p, q and r; show that z is the zero polynomial and distinguish it from r; compute p+q and p·q from coefficients and find their degrees; finally explain why p(1)=t(1) does not give p=t.
Univariate polynomials over the rationals
derivation-mko-polynomial-01
Concept guide
Over R=ℤ/4ℤ take f=1+2x and g=2x. Using the coefficient-sequence definition, compute and normalize f+g, f·g, f·f and g·g; prove the top-coefficient formula (fg)_{m+n}=f_m g_n and explain why deg(fg)=deg f+deg g fails here; finally prove that R[x] is not a finite ring.
Polynomial ring over a finite commutative ring
derivation-mko-polynomial-ring-over-finite-commutative-ring-01