Remainder Theorem

Q. Let $r\left(x\right)$ be the remainder when the polynomial $x^{135}+x^{125}–x^{115}+x^5+1$ is divided by $x^3–x$. Then

1. $r\left(x\right)$ is a nonzero constant
2. $r\left(x\right)$ is the zero polynomial
3. degree of $r\left(x\right)$ is one
4. degree of $r\left(x\right)$ is two

Q.

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

Q.

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

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

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

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

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

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

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

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

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

Q. Let $f\left(x\right)=x^5+a{x}^3+bx.$ The remainder when $f\left(x\right)$ is divided by $x+1$ is $-3.$ Then the remainder when $f\left(x\right)$ is divided by $x^2-1,$ is

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

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

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. Find the remainder when $x^3+4x^2-7x+6$ is divided by $x-1.$

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

Q. The remainder when $x^3+4x^2-7x+6$ is divided by $x-1$ is

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

Q. Find the remainder when $x^3+4x^2-7x+6$ is divided by $x-1.$

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

Q. Let $x^2+x+1$ is divisible by 3. If $x$ is divided by 3, the remainder will be

1. 2
2. 0
3. None of these
4. 1

Q. When a polynomial $p\left(x\right)$ is divided by $x-2,$ the remainder is $7.$ When $p\left(x\right)$ is divided by $x-3,$ the remainder is $9.$ If $r\left(x\right)$ is the remainder when $p\left(x\right)$ is divided by $(x-2)(x-3),$ then the value of $r(-1)$ is

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

Q. Let $f\left(x\right)=x^5+a{x}^3+bx.$ The remainder when $f\left(x\right)$ is divided by $x+1$ is $-3.$ Then the remainder when $f\left(x\right)$ is divided by $x^2-1,$ is

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

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

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

Q. Polynomial $P\left(x\right)$ contains only terms of odd degree. When $P\left(x\right)$ is divided by $(x-3)$, the remainder is $6$. If $P\left(x\right)$ is divided by $(x^2-9)$, then the remainder is $g\left(x\right)$. Then the value of $g\left(2\right)$ is