Calculating two unknown coefficients of a cubic expression when given two of the expression's linear factors.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "If $\\simplify{{a1}x+{b1}}$ and $\\simplify{{a2}x+{b2}}$ are factors of \\[\\simplify{{a1*a2*a3}x^3+m*x^2+n*x+{b1*b2*b3}},\\]
\nfind the values of $m$ and $n$.
", "advice": "In order to find the values of $m$ and $n$, we want to find the remaining factor of the polynomial, which can be done without knowing these two terms.
\nLet us first write the polynomial as a product of the factors we already know, and the unknown factor $(ax+b)$:
\n\\[\\simplify{{a1*a2*a3}x^3+m*x^2+n*x+{b1*b2*b3}} =(ax+b)(\\simplify{{a1}x+{b1}})(\\simplify{{a2}x+{b2}}).\\]
\nNow, the coefficient of the $x^3$ term is the product of the three $x$-coefficients from the linear factors: \\[ \\begin{split} \\simplify{{a1*a2*a3}} &\\,= \\simplify{{a1*a2}a} \\\\ \\ \\implies \\quad a &\\,=\\var{a3}.\\end{split}\\]
\nSimilarly, the constant term is the product of the three constants from the linear factors: \\[ \\begin{split} \\simplify{{b1*b2*b3}} &\\,= \\simplify{{b1*b2}b} \\\\ \\implies \\quad b &\\,=\\var{b3}. \\end{split} \\]
\nTherefore,
\n\\[\\simplify{{a1*a2*a3}x^3+m*x^2+n*x+{b1*b2*b3}} =(\\simplify{{a3}x+{b3}})(\\simplify{{a1}x+{b1}})(\\simplify{{a2}x+{b2}}).\\]
\nIf we now expand the right-hand side of this equation, we can compare coefficients to find the values of $m$ and $m$:
\n\\[ \\begin{split} \\simplify{{a1*a2*a3}x^3+m*x^2+n*x+{b1*b2*b3}} &\\,=(\\simplify{{a3}x+{b3}})(\\simplify{{a1}x+{b1}})(\\simplify{{a2}x+{b2}}) \\\\ &\\,=(\\simplify{{a3}x+{b3}})(\\simplify{{a1*a2}x^2+{a1*b2+a2*b1}x+{b1*b2}}) \\\\ &\\,= \\simplify[all,!collectNumbers,!cancelTerms]{{a3}x({a1*a2}x^2+{a1*b2+a2*b1}x+{b1*b2})+{b3}({a1*a2}x^2+{a1*b2+a2*b1}x+{b1*b2})} \\\\ &\\,=\\simplify[all,!collectNumbers,!cancelTerms]{{a3*a1*a2}x^3+{a3*a1*b2+a3*a2*b1}x^2+{a3*b1*b2} + {b3*a1*a2}x^2+{b3*a1*b2+b3*a2*b1}x+{b3*b1*b2}} \\\\ &\\,= \\simplify{{a1*a2*a3}x^3+{a1*a2*b3+a1*b2*a3+b1*a2*a3}x^2+{a1*b2*b3+b1*a2*b3+b1*b2*a3}x+{b1*b2*b3}}. \\end{split}\\]
\nTherefore,
\n\\[m=\\simplify{{a1*a2*b3+a1*b2*a3+b1*a2*a3}},\\quad n=\\simplify{{a1*b2*b3+b1*a2*b3+b1*b2*a3}}.\\]
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\n$n$ = [[1]]
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