Given $\\displaystyle f(z) = \\simplify[std]{z ^ 4+ {( -2) * r12}*z ^ 3+ {mz1+mz2+4*re(z1)*re(z2)} * z^2 -{2*(re(z2)*mz1+re(z1)*mz2)}z+{mz1*mz2}}$, one of the following complex numbers is a root $z_1$ of the equation $f(z)=0$.
\nChoose the correct value for $z_1$:[[0]]
", "stepsPenalty": 0, "scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["$\\simplify[std]{{a1}+{b1}i}$
", "$\\simplify[std]{{z3}}$
"], "showCorrectAnswer": true, "displayColumns": 4, "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": true, "scripts": {}, "maxMarks": 0, "type": "1_n_2", "minMarks": 0, "variableReplacementStrategy": "originalfirst", "matrix": [1, 0], "marks": 0}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"showCorrectAnswer": true, "scripts": {}, "type": "information", "variableReplacementStrategy": "originalfirst", "prompt": "Since you are given that $f(z)$ has a complex root $z_1$ and since $f(z)$ is a polynomial with real coefficients then the complex conjugate $\\overline{z_1}$ must also be a root.
\nSince $(z-z_1)(z-\\overline{z_1})=(z^2-2\\operatorname{Re}(z)+|z_1|^2)$ we have that:\\[f(z)=(z^2-2\\operatorname{Re}(z)+|z_1|^2)(z^2+az+b)=\\simplify{z ^ 4+ {( -2) * r12}*z ^ 3+ {mz1+mz2+4*re(z1)*re(z2)} * z^2 -{2*(re(z2)*mz1+re(z1)*mz2)}z+{mz1*mz2}}\\] where $a$ and $b$ are real.
\nLooking at the constant term we see that :
\n\\[|z_1|^2b = \\var{mz1*mz2}\\]
\nHence $|z_1|^2$ divides $ \\var{mz1*mz2}$.
\nAn easy test to see if one of the complex numbers given is not a root is to see if its modulus squared does not divide $ \\var{mz1*mz2}$. If it does not divide then the other must be the root.
", "variableReplacements": [], "marks": 0}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "Write down the quadratic factor with real coefficients, $q_1(z)$, of $f(z)$ which has $z_1$ as a root:
\n$q_1(z)=\\;$[[0]]
\nApart from $z_1$, $q_1(z)$ has another root $z_2$, which is also a root of $f(z)$.
\n$z_2=\\;$[[1]]
", "stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "z^2-{2*a1}*z+{mz1}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "{a1}-{b1}i", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"showCorrectAnswer": true, "scripts": {}, "type": "information", "variableReplacementStrategy": "originalfirst", "prompt": "If $z_1$ is a root then its conjugate $z_2$= $\\overline{z_1}$ is also a root.
", "variableReplacements": [], "marks": 0}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "Since $q_1(z)$ is a factor of $f(z)$ the other roots are given by finding the other quadratic factor $q_2(z)$ of $f(z)=q_1(z)q_2(z)$
\n$q_2(z)\\;=$[[0]]
\nFind the roots of $q_2(z)$ and hence the remaining two roots $z_3,\\;z_4$ of $f(z)$
\n$z_3=\\;$[[1]] (imaginary part negative)
\n$z_4=\\;$[[2]] (imaginary part positive).
", "stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "z^2-{2*a2}*z+{mz2}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "{a2}-{b2}*i", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "{a2}+{b2}*i", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"showCorrectAnswer": true, "scripts": {}, "type": "information", "variableReplacementStrategy": "originalfirst", "prompt": "$q_1(z)q_2(z)=f(z)$.
\nOnce you have found $q_1(z)$ then the easiest way to find $q_2(z)$ is to compare the terms in $z^3$ and the constant terms.
", "variableReplacements": [], "marks": 0}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "Given two complex numbers, find by inspection the one that is a root of a given quartic real polynomial $f(z)$ and hence find the other roots.
", "tags": ["checked2015", "MAS1602", "mas1602"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "15/07/2015:
\nAdded tags.
\n25/08/2012:
\nCopied question finding roots of a cubic in order to create new question finding roots of a quartic with 4 complex roots.
\nFunction ch finds the imaginary part of the complex number $z_3$ and ensures that $z_3$ is not a solution by insisting that $|z_3|^2$ does not divide the constant term of the polynomial. This is a simple way for the students to test to see which one of $z_1$ and $z_2$ is a solution.
\nAdded tags.
\nAdded description.
\nChecked calculation.OK.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Given two complex numbers, find by inspection the one that is a root of a given quartic real polynomial and hence find the other roots.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a) We use the method given in Show steps for part a).
\nNote that $|\\var{z1}|^2=\\var{mz1}$ divides the constant term $\\var{mz1*mz2}$,
\nbut that $|\\var{z3}|^2=\\var{mz3}$ does not divides the constant term $\\var{mz1*mz2}$.
\nHence $\\var{z1}$ is the root we are looking for.
\nb) A quadratic factor of $f(z)$.
\nSince $f(z)$ is a polynomial with real coefficients then if $z=z_1$ is a root we have that the conjugate $z=\\overline{z_1}$ is also a root.
\nHence the complex number $z_2=\\overline{\\var{z1}}=\\var{conj(z1)}$ is a root.
\nHence $q_1(z) = (z-(\\var{z1}))(z-(\\var{conj(z1)}))=\\simplify[std]{z^2-{2*a1}*z+{abs(z1)^2}}$ is a factor of $f(z)$.
\nc)The other quadratic factor and the other roots.
\nWe have that $f(z)=q_1(z)q_2(z)$, where $q_1(z)$ is as above and we have to find the quadratic $q_2(z)=z^2+az+b$ with real coefficients $a$ and $b$.
\n\\[\\begin{eqnarray*}f(z) &=& \\simplify[std]{z ^ 4+ {( -2) * r12}*z ^ 3+ {mz1+mz2+4*re(z1)*re(z2)} * z^2 -{2*(re(z2)*mz1+re(z1)*mz2)}z+{mz1*mz2}}\\\\&=&q_1(z)q_2(z)\\\\&=&(\\simplify[std]{z^2-{2*a1}*z+{mz1}})(z^2+az+b)\\\\&=&\\simplify[std]{z^4+(a-{2*a1})z^3+(b-{2*a1}*a+{mz1})*z^2+({mz1}a-{2*a1}b)*z+{mz1}*b}\\end{eqnarray*}\\]
\nIdentifying the constant terms and the coefficients of $z^3$ on both sides of this equation gives:
\n$a=\\var{-2*a2},\\;\\;b=\\var{mz2}$
\nHence $q_2(z)=\\simplify[std]{z^2-{2*a2}*z+{mz2}}$
\nYou can then find the roots of this quadratic, giving the other roots of $f(z)$:
\n$z_3=\\simplify[std]{{a2}-{b2}*i}$ (negative imaginary part)
\n$z_4=\\simplify[std]{{a2}+{b2}*i}$ (positive imaginary part)
\n", "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/"}]}