Question Number 135624 by liberty last updated on 14/Mar/21
$$ \\ $$ A polynomial p(x) leaves a remainder of -1 when divided by x - 1, a remainder of 3 when divided by x - 2, and a remainder of 4 when divided by x + 3. Let r(x) be the remainder when p(x) is divided by (x - 1) (x - 2) (x + 3). What is r(6)?\\n
Answered by EDWIN88 last updated on 14/Mar/21
$$\mathrm{Let}\:\mathrm{\color{mathred}{p}}\color{mathred}{\left(}\mathrm{\color{mathred}{x}}\color{mathred}{\right)}\color{mathred}{=}\color{mathred}{\left(}\mathrm{\color{mathred}{x}}\color{mathred}{−}\mathrm{\color{mathred}{1}}\color{mathred}{\right)}\color{mathred}{\left(}\mathrm{\color{mathred}{x}}\color{mathred}{−}\mathrm{\color{mathred}{2}}\color{mathred}{\right)}\color{mathred}{\left(}\mathrm{\color{mathred}{x}}\color{mathred}{+}\mathrm{\color{mathred}{3}}\color{mathred}{\right)}\color{mathred}{+}\mathrm{\color{mathred}{r}}\color{mathred}{\left(}\mathrm{\color{mathred}{x}}\color{mathred}{\right)} \\ $$ $$\mathrm{\color{mathred}{w}\color{mathred}{i}\color{mathred}{t}\color{mathred}{h}}\color{mathred}{\:}\mathrm{\color{mathred}{d}\color{mathred}{e}\color{mathred}{g}\color{mathred}{r}\color{mathred}{e}\color{mathred}{e}}\color{mathred}{\:}\mathrm{\color{mathred}{o}\color{mathred}{f}}\color{mathred}{\:}\mathrm{\color{mathred}{r}}\color{mathred}{\left(}\mathrm{\color{mathred}{x}}\color{mathred}{\right)}\mathrm{\color{mathred}{i}\color{mathred}{s}}\color{mathred}{\:}\mathrm{\color{mathred}{2}} \\ $$ $$\mathrm{\color{mathred}{a}\color{mathred}{n}\color{mathred}{d}}\color{mathred}{\:}\mathrm{\color{mathblue}{r}}\color{mathblue}{\left(}\mathrm{\color{mathblue}{1}}\color{mathblue}{\right)}\color{mathblue}{=}\color{mathblue}{−}\mathrm{\color{mathblue}{1}}\color{mathblue}{\:}\color{mathblue}{;}\color{mathblue}{\:}\mathrm{\color{mathblue}{r}}\color{mathblue}{\left(}\mathrm{\color{mathblue}{2}}\color{mathblue}{\right)}\color{mathblue}{=}\mathrm{\color{mathblue}{3}}\color{mathblue}{\:}\mathrm{\color{mathblue}{a}\color{mathblue}{n}\color{mathblue}{d}}\color{mathblue}{\:}\mathrm{\color{mathblue}{r}}\color{mathblue}{\left(}\color{mathblue}{−}\mathrm{\color{mathblue}{3}}\color{mathblue}{\right)}\color{mathblue}{=}\mathrm{\color{mathblue}{4}} \\ $$ $$\mathrm{\color{mathblue}{u}\color{mathblue}{s}\color{mathblue}{e}}\color{mathblue}{\:}\mathrm{\color{mathblue}{R}\color{mathblue}{e}\color{mathblue}{m}\color{mathblue}{a}\color{mathblue}{i}\color{mathblue}{n}\color{mathblue}{d}\color{mathblue}{e}\color{mathblue}{r}}\color{mathblue}{\:}\mathrm{\color{mathblue}{T}\color{mathblue}{h}\color{mathblue}{e}\color{mathblue}{o}\color{mathblue}{r}\color{mathblue}{e}\color{mathblue}{m}}\color{mathblue}{\:} \\ $$ $$\mathrm{\color{mathblue}{w}\color{mathblue}{e}}\color{mathblue}{\:}\mathrm{\color{mathblue}{g}\color{mathblue}{e}\color{mathblue}{t}}\color{mathblue}{\:}\mathrm{\color{mathblue}{r}}\color{mathblue}{\left(}\mathrm{\color{mathblue}{x}}\color{mathblue}{\right)}\color{mathblue}{=}\frac{\color{mathblue}{\left(}\mathrm{\color{mathblue}{x}}\color{mathblue}{−}\mathrm{\color{mathblue}{2}}\color{mathblue}{\right)}\color{mathblue}{\left(}\mathrm{\color{mathblue}{x}}\color{mathblue}{+}\mathrm{\color{mathblue}{3}}\color{mathblue}{\right)}}{\color{mathblue}{\left(}\mathrm{\color{mathblue}{1}}\color{mathblue}{−}\mathrm{\color{mathblue}{2}}\color{mathblue}{\right)}\color{mathblue}{\left(}\mathrm{\color{mathblue}{1}}\color{mathblue}{+}\mathrm{\color{mathblue}{3}}\color{mathblue}{\right)}}\color{mathblue}{.}\color{mathblue}{\left(}\color{mathblue}{−}\mathrm{\color{mathblue}{1}}\color{mathblue}{\right)}\color{mathblue}{+}\frac{\color{mathblue}{\left(}\mathrm{\color{mathblue}{x}}\color{mathblue}{−}\mathrm{\color{mathblue}{1}}\color{mathblue}{\right)}\color{mathblue}{\left(}\mathrm{\color{mathblue}{x}}\color{mathblue}{+}\mathrm{\color{mathblue}{3}}\color{mathblue}{\right)}}{\color{mathblue}{\left(}\mathrm{\color{mathblue}{2}}\color{mathblue}{−}\mathrm{\color{mathblue}{1}}\color{mathblue}{\right)}\color{mathblue}{\left(}\mathrm{\color{mathblue}{2}}\color{mathblue}{+}\mathrm{\color{mathblue}{3}}\color{mathblue}{\right)}}\color{mathblue}{.}\color{mathblue}{\left(}\mathrm{\color{mathblue}{3}}\color{mathblue}{\right)} \\ $$ $$\color{mathblue}{+}\color{mathblue}{\:}\frac{\color{mathblue}{\left(}\mathrm{\color{mathblue}{x}}\color{mathblue}{−}\mathrm{\color{mathblue}{1}}\color{mathblue}{\right)}\color{mathblue}{\left(}\mathrm{\color{mathblue}{x}}\color{mathblue}{−}\mathrm{\color{mathblue}{2}}\color{mathblue}{\right)}}{\color{mathblue}{\left(}\color{mathblue}{−}\mathrm{\color{mathblue}{3}}\color{mathblue}{−}\mathrm{\color{mathblue}{1}}\color{mathblue}{\right)}\color{mathblue}{\left(}\color{mathblue}{−}\mathrm{\color{mathblue}{3}}\color{mathblue}{−}\mathrm{\color{mathblue}{2}}\color{mathblue}{\right)}}\color{mathblue}{.}\color{mathblue}{\left(}\mathrm{\color{mathblue}{4}}\color{mathblue}{\right)} \\ $$ $$\mathrm{\color{mathblue}{t}\color{mathblue}{h}\color{mathblue}{e}\color{mathblue}{r}\color{mathblue}{e}\color{mathblue}{f}\color{mathblue}{o}\color{mathblue}{r}\color{mathblue}{e}}\color{mathblue}{\:}\mathrm{\color{mathred}{r}}\color{mathred}{\left(}\mathrm{\color{mathred}{6}}\color{mathred}{\right)}\color{mathred}{=}\frac{\color{mathred}{\left(}\mathrm{\color{mathred}{4}}\color{mathred}{\right)}\color{mathred}{\left(}\mathrm{\color{mathred}{9}}\color{mathred}{\right)}}{\color{mathred}{−}\mathrm{\color{mathred}{4}}}\color{mathred}{.}\color{mathred}{\left(}\color{mathred}{−}\mathrm{\color{mathred}{1}}\color{mathred}{\right)}\color{mathred}{+}\frac{\color{mathred}{\left(}\mathrm{\color{mathred}{5}}\color{mathred}{\right)}\color{mathred}{\left(}\mathrm{\color{mathred}{9}}\color{mathred}{\right)}}{\mathrm{\color{mathred}{5}}}\color{mathred}{.}\color{mathred}{\left(}\mathrm{\color{mathred}{3}}\color{mathred}{\right)}\color{mathred}{+}\frac{\color{mathred}{\left(}\mathrm{\color{mathred}{5}}\color{mathred}{\right)}\color{mathred}{\left(}\mathrm{\color{mathred}{4}}\color{mathred}{\right)}}{\color{mathred}{\left(}\color{mathred}{−}\mathrm{\color{mathred}{4}}\color{mathred}{\right)}\color{mathred}{\left(}\color{mathred}{−}\mathrm{\color{mathred}{5}}\color{mathred}{\right)}}\color{mathred}{.}\color{mathred}{\left(}\mathrm{\color{mathred}{4}}\color{mathred}{\right)} \\ $$ $$\color{mathred}{=}\color{mathred}{\:}\mathrm{\color{mathred}{9}}\color{mathred}{\:}\color{mathred}{+}\color{mathred}{\:}\mathrm{\color{mathred}{2}\color{mathred}{7}}\color{mathred}{\:}\color{mathred}{+}\color{mathred}{\:}\mathrm{\color{mathred}{4}}\color{mathred}{\:}\color{mathred}{=}\color{mathred}{\:}\mathrm{\color{mathred}{4}\color{mathred}{0}} \\ $$