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given-that-the-polynomial-p-x-3x-2-x-1-q-x-2x-4-of-degree-3-is-exactly-divisible-by-x-2-and-that-p-1-12-find-q-x-




Question Number 97606 by hardylanes last updated on 08/Jun/20
given that the polynomial p(x)=(3x+2)(x−1)q(x)−2x−4  of degree 3 is exactly divisible by x−2 and   that p(−1)=−12. find q(x).
$${given}\:{that}\:{the}\:{polynomial}\:{p}\left({x}\right)=\left(\mathrm{3}{x}+\mathrm{2}\right)\left({x}−\mathrm{1}\right){q}\left({x}\right)−\mathrm{2}{x}−\mathrm{4} \\ $$$${of}\:{degree}\:\mathrm{3}\:{is}\:{exactly}\:{divisible}\:{by}\:{x}−\mathrm{2}\:{and}\: \\ $$$${that}\:{p}\left(−\mathrm{1}\right)=−\mathrm{12}.\:{find}\:{q}\left({x}\right). \\ $$
Answered by mr W last updated on 08/Jun/20
g(x)=ax+b  p(−1)=(−3+2)(−1−1)g(−1)+2−4=−12  ⇒g(−1)=−5  ⇒−a+b=−5   ...(i)  p(2)=(6+2)(2−1)g(2)−4−4=0  ⇒g(2)=1  ⇒2a+b=1   ...(ii)  ⇒a=2  ⇒b=−3  ⇒g(x)=2x−3
$${g}\left({x}\right)={ax}+{b} \\ $$$${p}\left(−\mathrm{1}\right)=\left(−\mathrm{3}+\mathrm{2}\right)\left(−\mathrm{1}−\mathrm{1}\right){g}\left(−\mathrm{1}\right)+\mathrm{2}−\mathrm{4}=−\mathrm{12} \\ $$$$\Rightarrow{g}\left(−\mathrm{1}\right)=−\mathrm{5} \\ $$$$\Rightarrow−{a}+{b}=−\mathrm{5}\:\:\:…\left({i}\right) \\ $$$${p}\left(\mathrm{2}\right)=\left(\mathrm{6}+\mathrm{2}\right)\left(\mathrm{2}−\mathrm{1}\right){g}\left(\mathrm{2}\right)−\mathrm{4}−\mathrm{4}=\mathrm{0} \\ $$$$\Rightarrow{g}\left(\mathrm{2}\right)=\mathrm{1} \\ $$$$\Rightarrow\mathrm{2}{a}+{b}=\mathrm{1}\:\:\:…\left({ii}\right) \\ $$$$\Rightarrow{a}=\mathrm{2} \\ $$$$\Rightarrow{b}=−\mathrm{3} \\ $$$$\Rightarrow{g}\left({x}\right)=\mathrm{2}{x}−\mathrm{3} \\ $$

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