Factor, Remainder and Factorisation

Q.

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.

Factorise:
(i) $12x^2-7x+1$
(ii) $2x^2+7x+3$
(iii) $6x^2+5x-6$
(iv) $3x^2-x-4$

Q.

Which of the following are the factors of $x^2+3x+2$ ?

1. $(x+5)(x+2)$

2. $(x+3)(x+2)$

3. $(x+1)(x+2)$

4. $(x+2)(x+2)$

Q.

Quotient and remainder of a polynomial when divided by x+2 are 2x-1 and 3 respectively. Find the polynomial.

1. 

2. 

3. 

4. 

Q.

What is the factorization of $x^2-\frac{16x}{15}-1$?

1. (x-5)(x-)

2. (x-)(x+)

3. (x-)(x-)

4. (x-)(x+1)

Q.

The expression we get after factorising $2x^2+x-3$ is ___

1. $(2x+3)(x-2)$

2. $(2x+3)(x-1)$

3. $(5x+3)(x-1)$

4. $(2x+3)(8x-1)$

Q. Using Division Algorithm, match the following polynomials with their respective q(x) when divided by $x$.

1. 5x
2. 2x + 1
3. 3x + 2
4. 2x${}^2$ + 3

Q.

Question 1 (ii)
Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following: $p\left(x\right)=x^4-3x^2+4x+5,g\left(x\right)=x^2+1-x$

Q.

Which of the following are the factors of $x^2+7x+12$ ?

1. $(x+2)(x+4)$

2. $(x+1)(x+4)$

3. $(x+3)(x+9)$

4. $(x+3)(x+4)$

Q.

What is the factorization of $x^2+7x-8$?

1. (x-8) (x+1)

2. (x+7) (x-2)

3. (x+5) (x-9)

4. (x+8) (x-1)

Q. If the polynomial $f\left(x\right)=a{x}^3+bx-c$ is divisible by the polynomial $g\left(x\right)=x^2+bx+c$, then $ab=?$

1. $-\frac{1}{c}$
2. -1
3. 1
4. $\frac{1}{c}$

Q.

If polynomial= $2x^3+3x^2-2x$ , Divisor= $x-1$ and quotient = $2x^2+5x+3$, what is the remainder ?

1. 3

2. 4

3. 6

4. 2

Q. If the polynomial $f\left(x\right)=a{x}^3+bx-c$ is divisible by the polynomial $g\left(x\right)=x^2+bx+c$, then $ab=?$

1. $-\frac{1}{c}$
2. -1
3. 1
4. $\frac{1}{c}$

Q.

The polynomial P(x) = 4$x^3$ + 7$x^2$– 5x + t, when divided by x + 1 gives 3 as the remainder. Find the value of ‘t’.

1. -3

2. -4

3. -5

4. -6

Q. If $p\left(x\right)=2x^3+3x^2+4x+3$ is divided by a polynomial ______, then the quotient is $2x^2+x+3$ and the remainder is 0.

1. $x^4+1$
2. $4x^5+2x^4+9$
3. $x-1$
4. $x+1$

Q.

What is the factorization of $x^2-5x+6$?

1. (x-2) (x-3)

2. (x-1) (x+5)

3. (x-6) (x+1)

4. (x+5) (x-3)

Q. Using Division Algorithm, match the following polynomials with their respective q(x) when divided by $x$.

1. 5x
2. 2x + 1
3. 3x + 2
4. 2x${}^2$ + 3

Q. Using Division Algorithm, match the following polynomials with their respective q(x) when divided by $x$.

1. 5x
2. 2x + 1
3. 3x + 2
4. 2x${}^2$ + 3

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.

If $x^g+\frac{1}{x^a}$ is a polynomial, what could be the value of $a$?

1. any integer

2. any negative integer

3. only 1

4. any positive integer

Q.

If $x+x^{-1}=11$, evaluate $x^2+x^{-2}$.

1. 119
2. 120
3. 121
4. 122

Q. How do you factor completely $x^2-7x+18$?

Q. If a polynomial $p\left(x\right)=x^2-4$ is divided by a linear polynomial $(x-2)$ then the remainder is

1. $2$
2. $0$
3. $-2$
4. $-8$