Remainder Theorem

Q.

If the polynomials $2x^3+a{x}^2+3x-5$ and $x^3+x^2-4x+a$ leave the same remainder when divided by $x-2$, Find the value of $a$.

Q. If the expression $a{x}^4+b{x}^3-x^2+2x+3$ has remainder $4x+3$ when divided by $x^2+x-2$, then which of the following is/are true?

1. $a=2$
2. $b=2$
3. $a=1$
4. $b=1$

Q.

The remainder obtained when the polynomial $p\left(x\right)$ is divided by $(b-ax)$ is

1. $p\left(-\frac{b}{a}\right)$

2. $p\left(\frac{a}{b}\right)$

3. $p\left(\frac{b}{a}\right)$

4. $p\left(-\frac{a}{b}\right)$

Q. A polynomial in $x$ of degree greater than $3,$ leaves remainders $2,1$ and $-1$ when divided by $(x-1),(x+2)\text{and}(x+1)$ respectively. What will be the remainder when it is divided by $(x-1)(x+2)(x+1).$

1. $\frac{7}{6}{x}^2-\frac{3}{2}x-\frac{2}{3}$
2. $\frac{3}{2}x-\frac{2}{3}$
3. $\frac{7}{6}{x}^2+\frac{3}{2}x-\frac{2}{3}$
4. $\frac{3}{2}x+\frac{2}{3}$

Q.

If a polynomial $2x^2-7x-1$ is divided by $(x+3)$, then the remainder is

Q. Q-1 Find the remainder when 10^6 i s divided by 143 ? Q-2 Find the remainder When 5^100 is divided by 31 ?

Q. Using remainder theorem, find the value of m if the polynomial ​
$\text{f(x)}={\text{x}}^3+{\text{5x}}^2-\text{mx}+6$ leaves a remainder $2\text{m}$ when divided by ($\text{x-1}$).

Q. The degree of the polynomial $f\left(x\right)=\left|\begin{array}{ccc}1& 4& 20\\ 1& -2& 5\\ 1& 2x& 5x^2\end{array}\right|$ is

1. $0$
2. $1$
3. $2$
4. $4$

Q. If $f\left(x\right)=\left|\begin{array}{ccc}1& x& x+1\\ 2x& x(x-1)& (x+1)x\\ 3x(x-1)& x(x-1)(x-2)& x(x-1)(x+1)\end{array}\right|,$ then
$f\left(500\right)$ is equal to

1. $0$
2. $1$
3. $500$
4. $-500$

Q. For the polynomial $f\left(x\right)=x^3-6x^2+11x-6$, which of the following is/are true?
(Check using Remainder Theorem)

1. On dividing $f\left(x\right)$ by $x+1$, the remainder obtained is $-24$
2. On dividing $f\left(x\right)$ by $x-2$, the remainder obtained is $0$
3. On dividing $f\left(x\right)$ by $x-2$, the remainder obtained is $1$
4. All of the above

Q. Find the values of p and q so that by $x^4+x^3+8x^2+px+q$ is divisible by $x^2+1$. [Hint: Perform long division and then put remainder = 0]

Q. What is the remainder when $9^1+9^2+9^3+....+9^8$ is divided by 6?

1. 3
2. 2
3. 0
4. 5

Q. if the remainder which the number 9^100 gives when divided by 8 is R, find 19R

Q. Find the value of m and n when the polynomial $\text{f(x)}={\text{x}}^3-2{\text{x}}^2+\text{mn}+\text{mn}$ has a factor $(\text{x}+2)$ and leaves a remainder 9 when divided by $(\text{x}+1)$.​

Q. Find the HCF of $p\left(a\right)=2a^2-7a+3$ and $q\left(a\right)=2a^2+5a-3$.

Q. The polynomials $a{x}^3+3x^2-3$ and $2x^3-5x+a$ when divided by (x -4) leaves remainders $R_1,$ & $R_2$ respectively then value of 'a' if $2R_1-R_2=0.$

1. $-\frac{18}{127}$
2. $\frac{17}{127}$
3. $\frac{18}{127}$
4. $-\frac{17}{127}$

Q.

The factors of $x^4+x^2+25$ are

1. $(x^2+3x+5)(x^2–3x+5)$

2. $(x^2+3x+5)(x^2+3x–5)$

3. $(x^2+x+5)(x^2–x+5)$

4. None of these.

Q. Let $P\left(x\right)=x^2+bx+c$, where $b$ and $c$ are integers. If $P\left(x\right)$ is a facter of both $x^4+6x^2+25$ and $3x^4+4x^2+28x+5$, then value of $P\left(1\right)$ is -

1. $8$
2. $10$
3. $4$
4. $12$

Q.

Find the remainder when $x^3-a{x}^2+6x-a$ is divisible by $\left(x-a\right)$.

Q. Prove that:
$\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2\sec A$

Q. If a polynomial $p\left(x\right)$ is divided by $(x-a)$, then the remainder obtained is $p\left(a\right).$

1. True
2. False

Q. The remainder when $4x^3+2x^2-5x+7$ is divided by $x-2$ is

1. $-8$
2. $-37$
3. $37$
4. $8$

Q. If the given polynomial $f\left(x\right)=x^4+a{x}^3-5x^2+7x-6,$ is divided by $x-3$ and leaves remainder as $-3$ then the value of $a$ is

Q.

If $x^2+px+1$ is a factor of the expression $a{x}^3+bx+c$, then

1. $a^2-c^2=ab$

2. $a^2-c^2=-ab$

3. $a^2+c^2=-ab$

4. $a^2+c^2-ab=0$

Q. $685693564852136$
What will be the difference of the second-last digit and last digit of the second lowest number after the positions of only the first and the second digits within each number are interchanged?

1. $3$
2. $4$
3. $9$
4. $6$

Q. Divide, Write the quotient and the remainder.
$(5x^2-3x^2)÷x^2$

Q. The remainder when $4x^3+2x^2-5x+7$ is divided by $x-2$ is

1. $-8$
2. $-37$
3. $37$
4. $8$

Q. What number should be subtracted from ${\text{x}}^2+\text{x}+1$ so that the resulting polynomial is exactly divisible by $\text{(x - 2)}$.​

Q. Let $f\left(x\right)=x^3+a{x}^2+bx+c$ and $g\left(x\right)=x^3+b{x}^2+cx+a,$ where $a,b,c$ are integers with $c\ne 0.$ Suppose that the following conditions hold:
(a) $f\left(1\right)=0;$
(b) the roots of $g\left(x\right)=0$ are the squares of the roots of $f\left(x\right)=0.$
Find the value of $a^{2013}+b^{2013}+c^{2013}$.

1. 1
2. 0
3. -1
4. 2

Q. Let $P\left(x\right)$ be a polynomial, which when divided by $x-3$ and $x-5$ leaves remainders $10$ and $6$ respectively. If the polynomial is divided by $(x-3)(x-5)$ then the remainder is

1. $16$
2. $2x-16$
3. $-2x+16$
4. $60$