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"X^2+{g1}*X+{g0}", "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": "<p>Consider $f(X)$ and $g(X)$ as polynomials over &nbsp;$\\mathbb{Z}_{\\var{n}}$.</p>\n<p>Find their greatest common divisor (GCD) and enter it as a monic polynomial.</p>\n<p>Note that the coefficients all should be in $\\mathbb{Z}_{\\var{n}}=\\{0,\\;1,\\;2,\\;3,\\;4\\}$.</p>\n<p>GCD=[[0]].</p>\n<p></p>\n<p></p>\n<p>Click on Show steps to obtain information on monic polynomials over $\\mathbb{Z}_{\\var{n}}$.</p>", "steps": [{"type": "information", "prompt": "<p>Monic Polynomials.</p>\n<p></p>\n<p>A monic polynomial over $\\mathbb{Z}_{\\var{n}}$ &nbsp;has the coefficient of the greatest power equalling $1$.</p>\n<p>For example: in order to normalize &nbsp;$2X^2+3X+1$ over $\\mathbb{Z}_{\\var{n}}$ into a monic polynomial we are allowed to multiply the polynomial by a constant, and since$ 2^{-1} = 3 \\mod {\\var{n}}$ we have on multiplying the polynomial throughout by $3$ that</p>\n<p>$\\simplify{3 * (2 * X ^ 2 + 3 * X + 1) = 6 * X ^ 2 + 9 * X + 3 = X ^ 2 + 4 * X + 3}$ as a polynomial over $\\mathbb{Z}_{\\var{n}}$</p>\n<p>and which is now a monic polynomial .&nbsp;</p>\n<p></p>\n<p></p>", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "(((X ^ 2) + ({g1} * X) + {g0}) * ((X ^ 2) + ({h1} * X) + {h0}))", "musthave": {"message": "<p>Factorize the polynomial into irreducible factors.</p>", "showStrings": false, "partialCredit": 0, "strings": ["("]}, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "<p>Factorize the polynomial into irreducible factors.</p>", "showStrings": false, "partialCredit": 0, "strings": ["^4", "^3"]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>Find the irreducible factorization of $f(X)$ over &nbsp;$\\mathbb{Z}_{\\var{n}}$.</p>\n<p>$f(X)=\\;$[[0]]</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Let \\[\\begin{align}\\\\<br/>f(X)&amp;=\\simplify{ X ^ 4 + {f3} * X ^ 3 + {f2} * X ^ 2 + {f1} * X + {f0}}\\\\\\\\<br/>g(X)&amp;=\\simplify{X^5+{d4}*X^4+{d3}*X ^ 3 + {d2} * X ^ 2 + {d1} * X + {d0}}\\end{align}\\]</p>", "tags": ["checked2015", "factorize polynomials", "finite fields", "gcd", "irreducible polynomials", "MAS2213", "normalized polynomials", "polynomials over finite fields"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>$f(X)$ and $g(X)$ are polynomials over $\\mathbb{Z}_n$.</p>\n<p>Find their greatest common divisor (GCD) and enter it as a monic polynomial.</p>\n<p>Hence factorize $f(X)$ into irreducible factors.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>1. We have on using the Euclidean Algorithm that the GCD of $f(X)$ and $g(X)$ is $\\simplify{X^2+{g1}*X+{g0}}$ over $\\mathbb{Z}_{\\var{n}}$.</p>\n<p>Note that this GCD is a monic polynomial.</p>\n<p>2. Hence &nbsp;$\\simplify{X^2+{g1}*X+{g0}}$&nbsp; is a factor of $f(X)$ and is irreducible over $\\mathbb{Z}_{\\var{n}}$ as you can check.</p>\n<p>On dividing $f(X)$ by &nbsp;$\\simplify{X^2+{g1}*X+{g0}}$ we obtain:</p>\n<p>$f(X) = \\simplify{(X ^ 2 + {g1} * X + {g0}) * (X ^ 2 + {h1} * X + {h0})}$ over &nbsp;$\\mathbb{Z}_{\\var{n}}$.</p>\n<p>You can check that $\\simplify{X ^ 2 + {h1} * X + {h0}}$ is also irreducible over &nbsp;$\\mathbb{Z}_{\\var{n}}$.</p>\n<p>Hence the above factorization expresses $f(X)$ as the product of irreducible polynomials over &nbsp;$\\mathbb{Z}_{\\var{n}}$.</p>", "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/"}]}