Three parts, where the student has to collect together polynomials in $\\mathbb{Z}_3$, $\\mathbb{Z}_5$ and $\\mathbb{Z}_7$, respectively.
\nThe answer to part a has no $X$ term, because they cancel out.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Compute the following sums of polynomials.
", "advice": "Adding the polynomials we obtain $3X-\\var{a1+a2+a3}$ . But $3 \\equiv 0 \\pmod 3$ and $ -\\var{a1+a2+a3} \\equiv \\var{ans1} \\pmod 3$.
\nHence the sum is $\\var{ans1}$.
\nSumming up, we obtain:
\n\\begin{align}
(\\simplify{X ^ 6 + {b4} * X ^ 4 + {b2} * X ^ 2 + {b0}})&+\\simplify{ {g}*(X ^ 6 + {c5} * X ^ 5 + {c4} * X ^ 4 + {c3} * X ^ 3 + {c2} * X ^ 2 + {c1} * X + {c0})} \\\\
&= \\simplify{{1 + g} * X ^ 6 + {g * c5} * X ^ 5 + {b4 + g * c4} * X ^ 4 + {g * c3} * X ^ 3 + {b2 + g * c2} * X ^ 2 + {g * c1} * X + {b0 + g * c0}} \\\\
&\\equiv \\simplify{{d6} * X ^ 6 + {d5} * X ^ 5 + {d4} * X ^ 4 + {d3} * X ^ 3 + {d2} * X ^ 2 + {d1} * X + {d0}} \\pmod 5
\\end{align}
Once again, on summing up we obtain:
\n\\begin{align}
\\simplify{X ^ 6 + {f5} * X ^ 5 + {f4} * X ^ 4 + {f3} * X ^ 3 + {f2} * X ^ 2 + {f1} * X + {f0} } &+ \\simplify{ {h}*(X ^ 6 + {g5} * X ^ 5 + {g4} * X ^ 4 + {g3} * X ^ 3 + {g2} * X ^ 2 + {g1} * X + {g0})} \\\\
&= \\simplify{ {1 + h} * X ^ 6 + {f5 + h * g5} * X ^ 5 + {f4 + h * g4} * X ^ 4 + {f3 + h * g3} * X ^ 3 + {f2 + h * g2} * X ^ 2 + {f1 + h * g1} * X + {f0 + h * g0}} \\\\
&\\equiv \\simplify{{k6} * X ^ 6 + {k5} * X ^ 5 + {k4} * X ^ 4 + {k3} * X ^ 3 + {k2} * X ^ 2 + {k1} * X + {k0}} \\pmod 7
\\end{align}
Constant terms in the expression given in part a
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\n$(\\simplify{X ^ 6 + {f5}*X^5+{f4} * X ^ 4 +{f3} * X ^ 3 + {f2} * X ^ 2 +{f1}*X+ {f0}}) +\\simplify{ {h}(X ^ 6 + {g5} * X ^ 5 + {g4} * X ^ 4 + {g3} * X ^ 3 + {g2} * X ^ 2 + {g1} * X + {g0}) } = $ [[0]]
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