Find the following product of polynomials over $\\mathbb{Z}_3$.
\nAll coefficients of non zero terms in your answer should be $1$ or $2$.
\n$\\simplify{(X - { a1})(X - { a2})(X - { a3})} = $ [[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "((X ^ 3) + ({c22} * (X ^ 2)) + ({c21} * X) + {c20})", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Find the following product of polynomials over $\\mathbb{Z}_5$.
\nAll coefficients of non zero terms in your answer should be $1$, $2$, $3$ or $4$.
\n$\\simplify{(X -{ b1})(X -{ b2})(X - { b3}) } =$ [[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "((X ^ 3) + ({c32} * (X ^ 2)) + ({c31} * X) + {c30})", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Find the following product of polynomials over $\\mathbb{Z}_7$.
\nAll coefficients of non zero terms in your answer should be $1$, $2$, $3$, $4$, $5$ or $6$.
\n$\\simplify{(X - { c1})(X - { c2})(X - { c3}) } =$ [[0]]
\n", "showCorrectAnswer": true, "marks": 0}], "statement": "Compute the following products of polynomials.
", "tags": ["checked2015", "MAS2213"], "rulesets": {}, "preamble": {"css": "ul {\n padding-left: 2em;\n}", "js": ""}, "type": "question", "metadata": {"notes": "29/07/2013:
\n
Created version directly from the iassess question.
Expanding products of 3 linear polynomials over $\\mathbb{Z}_3,\\;\\mathbb{Z}_5,\\;\\mathbb{Z}_7$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Expanding out the polynomials gives:
\n\\[ \\simplify{(X + { -a1}) * (X + { -a2}) * (X + { -a3}) = X ^ 3 + {-a1-a2-a3} * X ^ 2 + {a1*a2+a1*a3+a2*a3} * X + {-a1*a2*a3}} \\]
\nBut we have to reduce the coefficients $\\bmod 3$ and we get:
\n\\begin{align}
\\text{Coefficient of } X^2 \\text{:} && \\var{-a1-a2-a3} &\\equiv \\var{c12} \\pmod 3 \\\\
\\text{Coefficient of } X \\text{:} && \\var{a2*a1+a2*a3+a1*a3} &\\equiv \\var{c11} \\pmod 3 \\\\
\\text{Constant term:} && \\var{-a1*a2*a3} &\\equiv \\var{c10} \\pmod 3 \\\\
\\end{align}
Hence the product is $\\simplify{((X ^ 3) + ({c12} * (X ^ 2)) + ({c11} * X) + {c10})}$.
\nExpanding out the polynomials gives:
\n\\[ \\simplify{(X + { -b1}) * (X + { -b2}) * (X + { -b3}) = X ^ 3 + {-b1-b2-b3} * X ^ 2 + {b1*b2+b1*b3+b2*b3} * X + {-b1*b2*b3}} \\]
\nBut we have to reduce the coefficients $\\bmod 5$ and we get:
\n\\begin{align}
\\text{Coefficient of } X^2 \\text{:} && \\var{-b1-b2-b3} &\\equiv \\var{c22} \\pmod 5 \\\\
\\text{Coefficient of } X \\text{:} && \\var{b2*b1+b2*b3+b1*b3} &\\equiv \\var{c21} \\pmod 5 \\\\
\\text{Constant term:} && \\var{-b1*b2*b3} &\\equiv \\var{c20} \\pmod 5 \\\\
\\end{align}
Hence the product is $\\simplify{((X ^ 3) + ({c22} * (X ^ 2)) + ({c21} * X) + {c20})}$.
\nExpanding out the polynomials gives:
\n\\[ \\simplify{(X + { -c1}) * (X + { -c2}) * (X + { -c3}) = X ^ 3 + {-c1-c2-c3} * X ^ 2 + {c1*c2+c1*c3+c2*c3} * X + {-c1*c2*c3}} \\]
\nBut we have to reduce the coefficients $\\operatorname{mod} 7$ and we get:
\n\\begin{align}
\\text{Coefficient of } X^2 \\text{:} && \\var{-c1-c2-c3} &\\equiv \\var{c32} \\pmod 7 \\\\
\\text{Coefficient of } X \\text{:} && \\var{c2*c1+c2*c3+c1*c3} &\\equiv \\var{c31} \\pmod 7 \\\\
\\text{Constant term:} && \\var{-c1*c2*c3} &\\equiv \\var{c30} \\pmod 7 \\\\
\\end{align}
Hence the product is $\\simplify{((X ^ 3) + ({c32} * (X ^ 2)) + ({c31} * X) + {c30})}$.
", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}