Long Division Method to Divide Two Polynomials

Q.

Verify the division algorithm for the polynomials
$p\left(x\right)=2x^4-6x^3+2x^2-x+2$ and $g\left(x\right)=x+2$.

Q.

If the polynomial $a{x}^3+b{x}^2-c$ is divisible by $x^2+bx+c$ then $ab$ is equal to?

Q. When $2^{256}$ is divided by 17, the remainder would be:

1. 1
2. 16
3. 14
4. 15

Q.

The remainder left out when $8^{2n}–{\left(62\right)}^{2n+1}$is divided by $9$, is

1. $0$

2. $2$

3. $7$

4. $8$

Q.

By actual division, find the quotient and the remainder when $(x^4+1)$ is divided by (x - 1).
Verify that remainder = f(1).

Q.

One factor of $x^4+x^2$ - 20 is $x^2$ +5, the other is

1. x + 4

2. x - 4

3. $x^2$ + 4

4. $x^2$ - 4

Q.

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.

$\text{Given the area of rectangle is}A=25a^2-35a+69.\text{The length is given}\text{as}(5a-3).\text{Find the width of the}\text{rectangle.}$

1. $5a-3$

2. $5a-4$

3. $4a-5$

4. $a-4$

Q. If $x^3-6x^2+ax+b$ is exactly divisible by $x^2-3x+2$, then $12a+22b=?$

1. 132
2. 264
3. 0
4. - 66

Q. $\text{Show that}(p-1)\text{is a factor of}{p}^{10}+p^8+p^6-p^4-p^2-1$

Q.

The quotient obtained when $6x^2$ is divided by 2x is:

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

Q.

If $4x^4–3x^3–3x^2+x–7$ is divided by $1–2x$ then the remainder will be:

1. $\frac{57}{8}$

2. $-\frac{59}{8}$

3. $\frac{55}{8}$

4. $-\frac{55}{8}$

Q.

Question 2
Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

Q.

Divide $6x^4-8x^3+12x-4$ by $2x^2$

1. $3x^2-4x+\frac{6}{x}-\frac{2}{x^2}$

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

3. $3x^2-4x+\frac{6}{5x}-\frac{2}{x^2}$

4. $3x^2-4x+\frac{6}{x}-\frac{1}{x^2}$

Q.

If $5x^3+5x^2-6x+9$ is divided by $(x+3)$, find the remainder.

Q. Question 1
Find the remainder when $x^3+3x^2+3x+1$ is divided by
(i) x + 1
(ii) $x-\frac{1}{2}$
(iii) x
(iv) $x+\pi$
(v) 5 + 2x

Q. On dividing $(5x^4+7x^3+2x^2–3x+5)$ by $(x+3)$, we get the quotient as

1. $8x^3+5x^2+26x–81$
2. $5x^3-8x^2+26x–81$
3. $5x^3+8x^2-26x+81$
4. $8x^3-5x^2-26x+81$

Q. Question 2
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 value of a. Also , find the remainder when p(x) is divided by x + 2.

Q.

$\text{Given the area of rectangle is}A=25a^2-35a+12.\text{The length is given}\text{as}(5a-3).\text{Find the width of the}\text{rectangle.}$

1. $5a-3$

2. $5a-4$

3. $4a-5$

4. $a-4$

Q.

When $p\left(x\right)=4x^3-12x^3+11x-5$ is divided by (2x - 1), the remainder is

1. 0

2. -5

3. -2

4. 2

Q.

Question 13
By actual division, find the quotient and the remainder when the first polynomial $x^4+1$ is divided by the polynomial (x - 1)

Q. Question 2
Find the remainder when $x^3-a{x}^2+6x-a$ is divided by x - a.

Q.

For the given statements select the correct option.

Assertion (A): If on dividing the polynomial $p\left(x\right)=x^2-3ax+3a-7$ by $(x+1)$, we get 6 as remainder, then $a=3$.

Reason (R): When a polynomial $p\left(x\right)$ is divided by $(x-a)$, then the remainder is $p\left(a\right)$.

Q.

Divide the following one and check by division algorithm.
$7772by58$

Q. Divide: $(3x^3+2x^2-1)by(x+2)$

Q.

Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively $\left(3mn,4np\right)$.

Q. What is meant by division algorithm give example?

Q. Divide $p\left(x\right)$ by $g\left(x\right)$ in the following case and verify division algorithm
$p\left(x\right)=x^2+4x+4;g\left(x\right)=x+2$

Q. By actual division, find the quotient and the remainder when the first polynomial $x^4+1$ is divided by the polynomial (x - 1)

Q. Determine which of the following polynomial has x – 2 a factor:
(i) $3x^2+6x-24$
(ii) $4x^2+x–2$