Question

Find the remainder when $p\left(x\right)=x^4-3x^3+2x^2+1$ is divided by $2x-1$.

• A. $\frac{2}{7}$
  
• B. $\frac{4}{9}$
  
• C. $\frac{19}{16}$
  
• D. $\frac{17}{19}$
  

Answer 1

The correct option is C $\frac{19}{16}$


Given that
$p\left(x\right)=x^4-3x^3+2x^2+1$
$2x-1=0\Rightarrow x=\frac{1}{2}$
By remainder theorem, when $p\left(x\right)$ is divided by $2x-1$, then remainder is given by $p\left(\frac{1}{2}\right)$
$p\left(\frac{1}{2}\right)={\left(\frac{1}{4}\right)}^4-3{\left(\frac{1}{2}\right)}^3+2{\left(\frac{1}{2}\right)}^2+1$
$=\frac{1}{16}-\frac{3}{8}+\frac{2}{4}+1$
$=\frac{1-6+8+16}{16}=\frac{19}{16}$
Remainder $=p\left(\frac{1}{2}\right)=\frac{19}{16}$
So, the correct answer is option (c).