Synthetic Division of Polynomials

Q.

Obtain all other zeroes of $3x^4+6x^3-2x^2-10x-5$, if two of its zeroes are $\sqrt{\frac{5}{3}}$ and $-\sqrt{\frac{5}{3}}$

Q.

When 15√15 is divided by 3√3, the quotient is:

1. 

2. 

3. 

4. 

Q.

If $p,q,r$ are simple propositions with truth values $T,F,T$ then the truth value $(~p\vee q)\wedge ~r\Rightarrow p$ is

1. True

2. False

3. True if r is false

4. True if q is true

Q.

If the sum of the coefficients of all even powers of in the product $(1+x+x^2+x^3\dots .+x^{2n})(1-x+x^2-x^3\dots .+x^{2n})$ is $61$, then $n$ is equal to:

Q.

Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: Area : 25$a^2$ – 35a + 12

Q.

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

Q.

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

Q.

Decide whether $(x+1)$ is a factor of the polynomial $(x^3+x^2-x-1)$ or not.

Q.

Which long division problem can be used to prove the formula for factoring the difference between two perfect cubes?

1. $a-b$ square root $a^2+ab+b^2$

2. $a+b$ square root $a^2-ab+b^2$

3. $a+b$ square root $a^3+0·a^2·b+0·a·b^2-b^3$

4. $a-b$ square root $a^3+0·a^2·b+0·a·b^2-b^3$

Q. Find the remainder when $x^3+3x^2+3x+1$ is divided by
$x-\frac{1}{2}$

Q.

If x and y are the two digits of the number 653xy such that this number is divisible by 80, then x + y = ______ .

1. 4

2. 8

3. 4 or 8

4. 6

Q. Find the remainder when $x^3+3x^2+3x+1$ is divided by
$5+2x$

Q. Find the quotient when the polynomial $x^3+x^2+1$ is divided by $x+2$ using synthetic division method.

1. $2x^2-3x+2$
2. $x^2-x+2$
3. $4x^2-x+2$
4. $x^2-x+6$

Q.

If the expression $a{x}^2+bx+c$ is equal to $4$, when $x=0$, leave the remainder $4$ when divided by $\left(x+1\right)$ and leaves a remainder $6$ when divided by $\left(x+2\right)$, then the values of $a$, $b$ and $c$ are respectively,

1. $1,1,4$

2. $2,2,4$

3. $3,3,4$

4. $4,4,4$

Q.

Is 2t+1 a factor of 4$t^3$+4$t^2$-t-1?

1. True

2. False

3. For t=7 only

4. Can't say

Q. Find the quotient when the polynomial $2x^3+2x^2+2$ is divided by $x-3$ using synthetic division method.

1. $7x^2+8x+24$
2. $2x^2+3x+24$
3. $2x^2+8x+24$
4. $2x^2+8x+26$

Q. Question 18
Show that p – 1 is a factor of $p^{10}–1$ and also of $p^{11}–1$.

Q. Find the quotient when the polynomial $4x^3+x^2-3$ is divided by $x-2$ using synthetic division method.

1. $4x^2+9x+18$
2. $4x^2+9x+22$
3. $5x^2+9x+18$
4. $4x^2+7x+18$

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

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

1. $a=6,b=8$ and $remainder=10$
2. $a=5,b=8$ and $remainder=10$
3. $a=5,b=7$ and $remainder=10$
4. None of these

Q. Find the quotient when the polynomial $x^2+1$ is divided by $x+2$ using synthetic division method.

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

Q. Two numbers x and y are such that when divided by 6 they leave remainders 4 and 5 respectively Find the remainder when $(x^2+y^2)$ is divided by 6

Q. Find the quotient when $6x^3+4x^2+2$ is divided by $x-2$ using synthetic division method.

1. $12x^2+16x+32$
2. $6x^2+16x+66$
3. $6x^2+16x+32$
4. $6x^2+16x+32$

Q. Divide the following polynomials by linear division method. Write the quotient and the remainder.
$(x^4+2x^3+3x^2+4x+5)÷(x+2)$

Q. Question 17
Determine which of the following polynomial has x – 2 a factor:
(i) $3x^2+6x-24$
(ii) $4x^2+x–2$

Q.

Divide $\left(3x^3+x^2+2x+5\right)$ by $\left(x^2+2x+1\right)$, find the quotient and remainder.

Q.

How do you find all the rational zeros of a polynomial function?

Q. Find the remainder when $x^3+3x^2+3x+1$ is divided by
$x+\pi$

Q. The remainder, when $({15}^{23}+{23}^{23})$ is divided by $19$, is

1. $4$
2. $23$
3. $0$
4. $17$

Q. A polynomial p(x) is divided by (2x - 1). The quotient and remainder obtained are $(7x^2+x+5)$ and 4 respectively. Find p(x).
OR
Find the quotient and remainder using synthetic division.
$(3x^3-2x^2+7x-5)÷(x+3)$.