Question

Apply the division algorithm to find the remainder on dividing $p\left(x\right)=x^4-3x^2+4x+5$ by $g\left(x\right)=x^2+1-x.$

• A. 8
• B. 6
• C. 4
• D. 2

Answer 1

The correct option is A 8

Given:
$p\left(x\right)=x^4-3x^2+4x+5$
$g\left(x\right)=x^2+1-x$
Degree of $q\left(x\right)=$ Degree of $p\left(x\right)-$Degree of $g\left(x\right)$
$=4-2=2$
Degree of $r\left(x\right)<$ Degree of $g\left(x\right)$
Let degree of $r\left(x\right)=1$
Let $q\left(x\right)=a{x}^2+bx+c$ and $r\left(x\right)=px+q$
By division algorithm
$p\left(x\right)=q\left(x\right)\times g\left(x\right)+r\left(x\right)$
$x^4-3x^2+4x+5=(a{x}^2+bx+c)(x^2+1-x)+(px+q)$
Comparing the coefficent of $x^4$.
$a=1.$
Comparing the coefficent of $x^3$.
$-a+b=0\Rightarrow b=a\Rightarrow b=1$
Comparing the coefficent of $x^2$
$a-b+c=-3$
$\Rightarrow 1-1+c=-3\Rightarrow c=-3$
Comparing the coefficent of x.
$b-c+p=4$
$\Rightarrow 1+3+p=4$
$\Rightarrow p=0$
Comparing the coefficent of constant term.
$c+q=5$
$-3+q=5$
$q=8.$
So, quotient $q\left(x\right)=a{x}^2+bx+c$
$=x^2+x-3$
and remainder $r\left(x\right)=px+q$
$=8$