Question

$\text{The polynomials}2x^3+a{x}^2+4x–12\text{and}{x}^3+x^2–2x+a\text{leave the same remainder}\text{when divided by}(x–3).\text{Find the value}\text{of}a.$

• A. $-1$
• B. $-2$
• C. $-3$
• D. $-4$

Answer 1

The correct option is C $-3$

$\text{Given polynomials:}p\left(x\right)=2x^3+a{x}^2+4x-12q\left(x\right)=x^3+x^2-2x+a$

$\text{Using remainder theorem, the}\text{remainders when}p\left(x\right)\text{and}q\left(x\right)\text{are divided by}(x-3)\text{are}p\left(3\right)\text{and}q\left(3\right)\text{respectively.}$

$p\left(3\right)=2\times {\left(3\right)}^3+a\times {\left(3\right)}^2+4\times 3-12p\left(3\right)=9a+54$

$q\left(3\right)={\left(3\right)}^3+{\left(3\right)}^2-2\times 3+a$
$q\left(3\right)=30+a$

$\text{Given: The remainder is the same.}\Rightarrow p\left(3\right)=q\left(3\right)\Rightarrow 9a+54=30+a\Rightarrow 8a=-24\Rightarrow a=-3$