if a²-2a-3 is a factor of a⁴+pa³+qa²+12a-9, find p and q.
Answers
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−9
a^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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