M L Aggarwal Solutions for Chapter: Polynomials, Exercise 9: Objective Questions

If on dividing the polynomial $f\left(x\right)=2x^4-3x^2-ax+b$ by $x-1$ and $x+1$ the remainders are $-3$ and $7$ respectively, then find the remainder when $f\left(x\right)$ is divided by $x-2$.

If $f\left(x\right)=a{x}^5+b{x}^3+cx+7$ where $a, b, c$ are real numbers, and$f\left(-3\right)=-3$, then find the value of $f\left(3\right)$.

When a polynomial $f\left(x\right)$ is divided by $\left(x-1\right)$, the remainder is $5$, and when it is divided by $\left(x-2\right)$, the remainder is $7$. Find the remainder when $f\left(x\right)$ is divided by $\left(x-1\right)\left(x-2\right)$.

If $\left(x^2-4\right)$ is a factor of the polynomial $a{x}^4+2x^3-3x^2+bx-4$, then find the values of $a$ and $b$. Hence, factorize the polynomial completely.

If the sum of two numbers is $7$ and the sum of their cubes is $133$, then find the sum of their squares.

If $x=\frac{1}{x-5}$, find the value of $x-\frac{1}{x}$.

If $x=\frac{1}{x-5}$, find the value of $x+\frac{1}{x}$.

If $x=\frac{1}{x-5}$, find the value of $x^3-\frac{1}{x^3}$.