Division Algorithm for a Polynomial

Q.

Factorize $x^2+5x+6$.

Q.

A number when divided by 61 gives 27 a quotient and 32 as remainder. Find the number.

Q.

Check whether $7+3x$ is a factor of $3x^3+7x$.

Q.

Question 4
On dividing $x^3-3x^2+x+2$ by a polynomial g(x), the quotient and remainder were $x-2$ and $-2x+4$, respectively. Find g(x).

Q.

Factorize: $6x^2-5x-6$

Q. Let p(x) be a polynomial of degree more than 2. When p(x) is divided by x-2, it leaves remainder 1 and when it is divided by x-3, it leaves remainder 3. Find the remainder, when p(x) is divided by (x-2)(x-3).

Q. If the polynomial $x^4+2x^3+8x^2+12x+18$ is divided by another polynomial $x^2+5,$ then the remainder comes out to be ax + b. The value of 2a + 3b is

1. 12
2. 11
3. 13
4. 18

Q.

Given that $x-\sqrt{5}$ is a factor of the cubic polynomial $x^3-3\sqrt{5}{x}^2+13x-3\sqrt{5}$ find all the zeroes of the polynomial.

Q. The polynomial which when divided by −x² + x − 1 gives a quotient x − 2 and remainder 3, is

(a) x³ − 3x² + 3x − 5
(b) −x³ − 3x² − 3x − 5
(c) −x³ + 3x² − 3x + 5
(d) x³ − 3x² − 3x + 5

Q. If the polynomial $f\left(x\right)=x^4-6x^3+16x^2-25x+10$ is divided by another polynomial $x^2-2x+k$, the remainder comes out to be $x+a$, find k and a.
[4 MARKS]

Q.

Find all the zeros of $(x^4+x^3-23x^2-3x+60)$, if it is given that two of its zeros are $\sqrt{3}and-\sqrt{3}.$

Q. Obtain all the zeroes of the polynomial $3x^4-12x^3+5x^2+16x-12,$ if two of its zeroes are $\frac{2}{\sqrt{3}}and\frac{-2}{\sqrt{3}}.$

1. 
   $\left(\frac{2}{\sqrt{3}}\right),(-\frac{2}{\sqrt{3}}),3,1$
2. 
   $\left(\frac{2}{\sqrt{3}}\right),(-\frac{2}{\sqrt{3}}),3,2$
3. 
   $\left(\frac{2}{\sqrt{3}}\right),(-\frac{2}{\sqrt{3}}),4,1$
4. 
   $\left(\frac{2}{\sqrt{3}}\right),(-\frac{2}{\sqrt{3}}),1,1$

Q.

Factorise $3x^3-x^2-3x+1$.

Q.

Question 1 (i)
Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:

$p\left(x\right)=x^3-3x^2+5x-3,g\left(x\right)=x^2-2$

Q. Find the zeroes of the polynomial $f\left(x\right)=x^3-8x^2+19x-12,\text{if it is given}\text{that the product of its two zereoes is 4.}$

1. 
   3, 4 and 2
2. 
   3, 4 and 1
3. 
   3, 2 and 1
4. 
   3, 1 and 7

Q.

Factorise $x^3+x–3x^2–3$.

Q.

Factorise $x^3-3x^2-9x-5$

Q.

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:
(i) p(x)=2$x^3$+$x^2$−2x−1, g(x)=x+1
(ii) p(x)=$x^3$+3$x^2$+3x+1, g(x)=x+2
(iii) p(x)=$x^3$−4$x^2$+x+6, g(x)=x−3

Q. If 3 and -3 are two zeros of the polynomial $(x^4+x^3-11x^2-9x+18)$, find all the zeros of the given polynomial.

Q.

What must be subtracted from polynomial $f\left(x\right)=x^4+2x^3-13x^2-12x+21$ so that the resulting polynomial is exactly divisible by $x^2-4x+3$ ?

Q.

The remainder when $x^{101}+101$ is divided by x+1 is:

1. 100

2. 101

3. 102

4. 103

Q.

On dividing a polynomial p(x) by a nonzero polynomial q(x), let g(x) be the quotient and r(x) be the remainder then p(x) = q(x).g(x)+r(x), where

(a) r(x) = 0 always

(b) deg r(x) < deg g(x) always

(c) either r(x) = 0 or deg r(x) < deg g(x)

(d) r(x) = g(x)

Q.

The polynomial $p\left(x\right)=x^4-2x^3+3x^2-ax+3a-7$ when divided by $x+1$ leaves the remainder $19$.

Find the values of $a$. Also find the remainder when $p\left(x\right)$ is divided by$x+2$.

Q.

Factorise: $4y^2-4y+1$

Q. Question 1 (i)
Answer the following and justify
Can $x^2–1$ be the quotient on division of $x6+2x^3+x-1$ by a polynomial in x of degree 5?

Q. If the polynomial $(x^4+2x^3+8x^2+12x+18)$ is divided by another polynomial $(x^2+5)$, the remainder comes out to be $(px+q)$, find the values of p and q.

Q. What must be added to $f\left(x\right)=4x^4+2x^3-2x^2+x-1$ so that the resulting polynomial is divisible by $g\left(x\right)=x^2+2x-3$
[4 MARKS]

Q.

Find the quatient and the remainder when\

$f\left(x\right)=x^4-3x^2+4x+5$ is divided by $g\left(x\right)=x^2+1-x.$

Q.

Given that $\sqrt{2}$ is a zero of the cubic polynomial $6x^3+\sqrt{2}{x}^2-10x-4\sqrt{2}$, find its other two zeroes.

Q. $\text{If 2 is one of the zeros of}{x}^3-3x^2-4x-12,\text{then find the remaining}\text{two zeroes.}$

1. $\text{-3 and 2}$
2. $\text{-3 and -2}$
3. $\text{3 and 2}$
4. $\text{3 and -2}$