M L Aggarwal Solutions for Chapter: Polynomials, Exercise 3: Exercise 2.3

Using Remainder theorem, check whether $p\left(x\right)$ is a multiple of $g\left(x\right)$ or not:

$p\left(x\right)=2x^3-11x^2-4x+5, g\left(x\right)=2x+1$

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

Divide $x^3+4x^2-3x-10$ by $\left(x+1\right)$ and verify the remainder obtained by Remainder theorem.

Find the quotient and remainder when $6x^4+11x^3+13x^2-3x+25$ is divided by $\left(3x+4\right)$. Also check the remainder by using Remainder theorem.

If the remainder on dividing the polynomial $2x^4-k{x}^2+5x-3k+3$ by $\left(x+2\right)$ is $4$, then find the value of $k$.

If the polynomials $a{x}^3+3x^2-13$ and $5x^3-8x+a$ leave the same remainder when divided by $x+1$, then find the value of $a$.

The polynomial $x^3+2x^2-5kx-7$ when divided by $\left(x+1\right)$ leaves the remainder $p$ and the polynomial $x^3+k{x}^2-12x+6$ when divided by $\left(x-2\right)$ leaves the remainder $q$. If $2p+q=6$, then find the value of $k$.

If $p\left(x\right)=x^3-a{x}^2+bx+3$ leaves a remainder $-19$ when divided by $\left(x+2\right)$ and remainder $17$ when divided by $\left(x-2\right)$, prove that $a+b=6$.