Division Algorithm for Polynomial

Q. Simplify
$\frac{1}{x+p}+\frac{1}{x+q}+\frac{1}{x+r}+\frac{px}{x^3+p{x}^2}+\frac{qx}{x^3+p{x}^2}+\frac{rx}{x^3+rx}$

1. $\frac{1}{x+p+q+r}$
2. $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}$
3. $\frac{3}{x}$
4. $\frac{3}{p+q+r}$

Q.

Simpson’s one-third rule is applied to a function for its

1. Expansion

2. Differentiation

3. Numerical integration

4. None of these

Q. When $x^4+x^3-2x^2+x+2$ is divided by 𝒈(𝒙), the quotient is 𝒙−𝟏 and the remainder is 𝟑. Find 𝒈(𝒙).

Q.

The polynomials $k{x}^3+4x^2+3x-4$ and $x^3-4x+k$ leave the same remainder when divided by $(x-3)$, find the value of $k$.

Q.

Divide the polynomial $\mathrm{p}\left(\mathrm{x}\right)$ by the polynomial $\mathrm{g}\left(\mathrm{x}\right)$ and find quotient and remainder for the following $p\left(x\right)=8x^3-6x+7;g\left(x\right)=2x-1$

Q.

Divide the polynomial $\mathrm{p}\left(\mathrm{x}\right)$ by the polynomial $\mathrm{g}\left(\mathrm{x}\right)$ and find quotient and remainder for the following:
$\mathrm{p}\left(\mathrm{x}\right)=4{\mathrm{x}}^3+5{\mathrm{x}}^2+6;\mathrm{g}\left(\mathrm{x}\right)=\mathrm{x}-2$