Question

The polynomial $2x^4-5x^3+2x^2-x+2$ is divisible by

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

Answer 1

Answer: (A), (D)

The correct options are
A $x^2-3x+2$
D $2x^2+x+1$

Let, $f\left(x\right)=2x^4-5x^3+2x^2-x+2$

By Trial and Error method,

$f\left(2\right)=2(2{)}^4-5(2{)}^3+2(2{)}^2-2+2$

$=2\times 16-5\times 8+2\times 4-2+2$

$=32-40+8-2+2=0$

$\therefore (x-2)$ is a factor of $f\left(x\right)$

Again, by Trial and Error method,

$f\left(1\right)=2(1{)}^4-5(1{)}^3+2(1{)}^2-1+2$

$=2-5+2-1+2=0$

$\therefore (x-1)$ is a factor of $f\left(x\right)$

Now, $(x-1)(x-2)=x^2-3x+2$

$\therefore x^2-3x+2$ is a factor of $f\left(x\right)$

To find the other factors, we divide $f\left(x\right)$ by $x^2-3x+2$