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If a²-2a-3 is a factor of a⁴+pa³+qa²+12a-9, find p and q. |
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Answer» Factor theorem:\text{(x-a) is a factor of F(x) iff f(a) =0}(x-a) is a factor of f(x) iff f(a) =0\textbf{Given:}Given:a^2-2a-3\;\text{is a factor of}\;a^4+P\,a^3+q\,a^2+12a-9a 2 −2a−3is a factor ofa 4 +pa 3 +qa 2 +12a−9\textbf{To find:}To find:\text{The value of $p^2-2q-3$}The value of p 2 −2q−3\textbf{Solution:}Solution:\text{LET}\;f(a)=a^4+p\,a^3+q\,a^2+12a-9Letf(a)=a 4 +pa 3 +qa 2 +12a−9a^2-2a-3=(a-3)(a+1)a 2 −2a−3=(a−3)(a+1)\textbf{Since (a-3) is a factor of f(a), we have f(3)=0}Since (a-3) is a factor of f(a), we have f(3)=0\implies\,3^4+p\,3^3+q\,3^2+12(3)-9=0⟹3 4 +P3 3 +q3 2 +12(3)−9=0\implies\,81+27\,p+9\,q+27=0⟹81+27p+9q+27=0\implies\,27\,p+ |
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