Division Algorithm for Polynomial

Q.

Divide $f\left(x\right)$ by $g\left(x\right)$ and find the quotient.
$f\left(x\right)=x^3-125,g\left(x\right)=x^2+5x+25$

1. x + 5

2. x - 5

3. x + 2

4. $x^2+5x$

Q.

$f\left(x\right)=\left[\frac{x(x-p)}{(q-p)}\right]+\left[\frac{x(x-q)}{(p-q)}\right],p\ne q.$ What is the value of $f\left(p\right)+f\left(q\right)$?

1. $f(p-q)$

2. $f(p+q)$

3. $f\left(p\right(p+q\left)\right)$

4. $f\left(q\right(p-q\left)\right)$

Q. If $a+b+c=9,$ and $a^2+b^2+c^2=35,$ find thevalue of $a^3+b^3+c^3-3abc.$

Q. What's the remainder when $x^3-a{x}^2+6x-a$ is divided by x - a.

Q.

Evaluate $\frac{a^3+b^3+c^3-3abc}{(a+b+c)},\text{where}(a+b+c)\ne 0$.

1. $a^2+b^2+c^2-ab-bc-ca$
2. $a^2+b^2+c^2+ab-bc-ca$
3. $a^2+b^2+c^2+ab+bc+ca$
4. $a^2-b^2-c^2+ab+bc+ca$
   

Q. If $P\left(x\right)=\frac{1+5x}{1-5x}\&Q\left(x\right)=\frac{1-5x}{1+5x},$ then $\frac{P\left(x\right)}{Q\left(x\right)}$ will be

1. 1
2. $\frac{1-25x^2}{1+25x^2}$
3. $\frac{(1+5x{)}^2}{(1-5x{)}^2}$
4. $\frac{1+25x^2}{1-25x^2}$

Q. When $x^2-4x+11$ is divided by $x-2$, find the sum of quotient and remainder.

1. x + 2
2. x + 5
3. x + 4
4. 2

Q.

Priya lost her homework paper on polynomials and she doesn't remember the divisor which, on dividing the polynomial $x^3-3x^2+x+2$ gives quotient $(x-2)$ and remainder $(-2x+4).$ Find the divisor.

1. $5x^2-7x+9$

2. $x^2-8x+7$

3. $x^2-x+1$

4. $6x^2-x+11$

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

Q. $\text{If}(x-3)\text{and}(x-2)\text{are two factors of}\text{the polynomial}{x}^3-6x^2+11x-6,\text{then find the third factor.}$

1. $x-1$
2. $x-4$
3. $x-5$
4. $x-6$

Q.

When $3x^3+2x^2-x+4$ is divided by $3+2x$ the remainder is $\frac{-1}{8}$.

1. True

2. False

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

1. $x^3+2x^2+x+1$
2. $x^3+2x^2+1$
3. $x^3+x^2+1$
4. $x^3+x^2+x$

Q.

The polynomial $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^4-2{\mathrm{x}}^3+3{\mathrm{x}}^2-\mathrm{ax}+\mathrm{b}$ when divided by $(\mathrm{x}-1)$ and $(\mathrm{x}+1)$ leaves remainder $5$and $19$. Find the values of $\mathrm{a}$ and $\mathrm{b}$. Hence, find the remainder when $\mathrm{f}\left(\mathrm{x}\right)$ is divided by $(\mathrm{x}-2)$.