Question

$\text{Find the polynomial from the given}\text{information.}$

$\text{Quotient}=x^3+2x$
$\text{Divisor}=x^2+\frac{1}{2}x$
$\text{Remainder}=5$

• A. $x^5+x^4+2x^3+x^2+5$
• B. $x^5+\frac{1}{2}{x}^4+2x^3+x^2$
• C. $x^5+\frac{1}{2}{x}^4+2x^3+x^2+5$
• D. $x^5+2x^3+x^2+5$

Answer 1

The correct option is C $x^5+\frac{1}{2}{x}^4+2x^3+x^2+5$

$\text{Let the polynomial be}p\left(x\right);\text{Divisor be}d\left(x\right);\text{Quotient be}q\left(x\right);\text{Remainder be}r\left(x\right).$

$\text{Given: Quotient,}q\left(x\right)=x^3+2x\text{Divisor,}d\left(x\right)=x^2+\frac{1}{2}x\text{Remainder,}r\left(x\right)=5$

$\text{We know that, by division algorithm,}$
$\text{Dividend = Divisor}\times \text{Quotient}+\text{Remainder}$

$\text{Then,}p\left(x\right)=d\left(x\right)\times q\left(x\right)+r\left(x\right)$

$p\left(x\right)=(x^2+\frac{1}{2}x)(x^3+2x)+5=x^2(x^3+2x)+\frac{1}{2}x(x^3+2x)+5=x^5+2x^3+\frac{1}{2}{x}^4+\frac{1}{2}\times 2x^2+5p\left(x\right)=x^5+2x^3+\frac{1}{2}{x}^4+x^2+5$

$\text{On rearranging, we get,}p\left(x\right)=x^5+\frac{1}{2}{x}^4+2x^3+x^2+5$