Answer: The correct option is
(C) [tex](7x^3-14x^2+28x-55)+\dfrac{124}{x+2}.[/tex]
Step-by-step explanation: We are given to select the correct form of [tex]q(x)+\dfrac{r(x)}{b(x)}[/tex] for the following expression :
[tex]E=\dfrac{7x^4+x+14}{x+2}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
To find the required expression, we try to divide the numerator by denominator till we get a constant remainder.
We have from (i) that
[tex]E\\\\\\=\dfrac{7x^4+x+14}{x+2}\\\\\\=\dfrac{7x^3(x+2)-14x^2(x+2)+28x(x+2)-55(x+2)+124}{x+2}\\\\\\=7x^3-14x^2+28x-55+\dfrac{124}{x+2}\\\\\\=q(x)+\dfrac{r(x)}{b(x)},[/tex]
where,
[tex]q(x)=7x^3-14x^2+28x-55,~~~~~r(x)=124,~~~~~b(x)=x+2.[/tex]
Thus, the required form is
[tex]\dfrac{7x^4+x+14}{x+2}=7x^3-14x^2+28x-55+\dfrac{124}{x+2}.[/tex]
Option (C) is CORRECT.
