Long Division Method to Divide Two Polynomials

Q.

How do you factor $6x^2+5x-6$?

Q.

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

$p\left(x\right)=x^4-5x+6,g\left(x\right)=2-x^2$

Q.

Using the remainder Theorem, factorise each of the following completely:
(i) $3x^3+2x^2-19x+6$
(ii)$2x^3+x^2-13x+6$
(iii) $3x^3+2x^2-23x-30$
(iv) $4x^3+7x^2-36x-63$
(v) $x^3+x^2-4x-4.$

Q.

If a polynomial of degree 8 is divided by polynomial of degree 5 then find the degree of the remainder

Q.

Find the factors of the expression $x^3-5x^2+8x-4$

1. (x - 1), (x - 2) and (x - 2)

2. (x - 9) (x - 2) (x - 3)

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

4. (x - 4) (x - 1) and (x - 3)

Q.

If $8^x=\frac{64}{2^x}$ , then $x=$ _____

Q.

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

1. True

2. False

Q.

A polynomial of degree 6 is divided by a polynomial of degree 2 if the divisor is the factor of the dividend find the degree of the remainder

Q.

Find the values of a and b such that the polynomial $x^3-a{x}^2$ - bx + 4 has (x - 1) and (x + 1) as factors

1. a = 3

   b = 4

2. a = 1

   b = 4

3. a = 5

   b = 2

4. a = 4

   b = 5

Q.

Find the quotient when $x^3+2x^2$ - x - 2 is divided by (x + 2)

1. $5x^2-4$

2. $x^2-1$

3. $2x^2-3$

4. $x^2+5$

Q.

If the degree of the divisor g(x) is one then the degree of the remainder r(x) is

1. 2

2. 1

3. 0

4. 3

Q.

What number should be added to $3x^3-5x^2+6x$ so that when resulting polynomial is divided by x - 3, the remainder is 8

Q.

Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

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.

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. Divide $x^3-125by{x}^2+5x+25$ and find the quotient.

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

Q.

In polynomial division , the degree of the remainder is always ___ ( greater / less ) than the degree of the divisor.

Q.

Find the remainder and quotient when $x^3+x^2$ - 9x - 9 is divided by (x + 1)

1. Quotient = $x^2$ - 9

   Remainder = 0

2. Quotient = $x^2$ - 9

   Remainder = 9

3. Quotient = $4x^2$ + 5

   Remainder = 5

4. Quotient = $9x^2$ + 4

   Remainder = 8

Q.

If dividend = $x^4+x^3-2x^2+x+1$ and divisor = $(x-1)$, find the quotient, $q\left(x\right)$ and remainder $r\left(x\right)$.

1. q(x) = $x^3+2x^2+x+1$ and r(x) = -1

2. q(x) = -$x^3+x^2+x$ and r(x) = 4

3. q(x) = $x^3+2x^2+1$ and r(x) = 2

4. q(x) = $x^3+2x^2+1$ and r(x) = 0

Q. What must be subtracted from $x^3-6x^2-15x+80$ so that the result is exactly divisible by $x^2+x-12$?

Q. Which one of the following is a polynomial?

1. $\frac{x^2}{2}-\frac{2}{x^2}$
2. $\sqrt{2x}-1$
3. $\frac{x-1}{x+1}$
4. $x^2+\frac{3x^{\frac{3}{2}}}{\sqrt{x}}$

Q. If the polynomial $x^4-6x^3+16x^2-25x+10$ is divided by $x^2-2x+k$ the remainder comes out to be $x+a$. Find the value of $a$.

Q. If $x^4+x^3-2x^2+x+1$ is divided by $x-1$ then remainder = .

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

Q.

If dividend = $x^3+13x^2+32x+20$, divisor = $x^2+11x+10$ remainder = 0. Find the quotient q(x)

1. x + 2

2. x – 2

3. x + 1

4. x– 1

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 values of $p$ and $q$.

1. $p=2q=3$
2. $p=2q=2$
3. $p=3q=2$
4. $p=3q=3$

Q.

$4x^3+4x^2-x-1$ is a multiple of-

1. 2x-1

2. 2x+1

3. x-1

4. x-4

Q.

$x^3+3x^2-4x-12$, when divided by (x - 2), leaves a remainder of $a{x}^2+bx+c$. Find a, b, c

1. a = 12 b = 4 c = 8

2. a = 4 b = 2 c = 8

3. b = 5 a = 1 c = 6

4. c = 8 a = 9 b = 4

Q. State whether True or False:

1. Yes
2. no
3. Invalid question
4. Cannot be determined

Q. Examine whether $x-2$ is a factor of $2x^3-3x^2-5x+6$ or not using factor theorem.

Q.

If $x^{51}+51$ is divided by $x+1$, then the remainder is

1. 1

2. 50

3. 49

4. 0