Question

Let $r(x)$ be the remainder when the polynomial $x^{135} + x^{125} - x^{115} + x^5 + 1$ is divided by $x^3 - x$. Then,

• A. $r(x)$ is the zero polynomial
• B. $r(x)$ is a non-zero constant
• C. degree of $r(x)$ is one
• D. degree of $r(x)$ is two

Answer 1

Answer: (C)

Let $p(x) = x^{135} + x^{125} - x^{115} + x^5 + 1$,
$q(x) = x^3 - x$ and $p(x) = q(x)k + r(x)$
$x^{135} + x^{125} - x^{115} + x^5 + 1$
$= (x^3 - x)k + ax^2 + 5x + c$
$[\because r(x) = ax^2 + bx + c]$
Put $x = 0$,
$\therefore c = 1$
Put $x = 1, 3 = a + b + c$
$\Rightarrow 3 = a + b + 1$
$\Rightarrow a + b = 2 \,... (i)$
Put $x = - 1, - 1 = a - b + c $
$\Rightarrow - 1 = a - b + 1$
$\Rightarrow a - b = - 2 \,... (ii)$
From Eqs. (i) and (ii), we get
$a = 0, b = 2$
$\therefore r(x) = 2x + 1$
$\therefore $ Degree of $r(x) = 1$