// Numbas version: exam_results_page_options {"name": "Polynomials: equating coefficients 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "
Suppose that during some working, you find that
\n{poly} $=$ {otherpoly}
\nfor all $x$.
", "tags": [], "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["k", "n", "poly", "otherpoly", "j"], "functions": {}, "variable_groups": [], "name": "Polynomials: equating coefficients 1", "preamble": {"js": "", "css": ""}, "advice": "Two polynomials are equal if and only if their corresponding coefficients are also equal.
\nSo we 'equate the coefficients' of the polynomials given in this question, this means that the constant terms must be equal, ie
\n\\[\\simplify{{k[0]}=C_0+{j[0]}}\\]
\nthe coefficients of the $x$ terms must be equal, ie
\n\\[\\simplify{{k[1]}={j[1]}C_1+{j[2]}}\\]
\nthe coefficients of the $x^2$ terms must be equal, ie
\n\\[\\simplify{{k[2]}=C_2}\\]
\nthe coefficients of the $x^3$ terms must be equal, ie
\n\\[\\simplify{{k[3]}={j[3]}C_3}\\]
\nthe coefficients of the $x^4$ terms must be equal, ie
\n\\[\\simplify{{k[4]}=C_4/{j[4]}}\\]
\nand the coefficients of the $x^5$ terms must be equal, ie
\n\\[\\simplify{{k[5]}=C_5/{j[5]}+{j[6]}}.\\]
\nSolving equal of these equations for $C_i$ gives our required values.
", "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "Equating coefficients of a polynomial. Basic ones that don't require simultaneous equations.
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\n$C_0=$ [[0]], $C_1=$ [[1]], $C_2=$ [[2]], $C_3=$ [[3]], $C_4=$ [[4]], $C_5=$ [[5]]
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