Embibe Experts Solutions for Chapter: Algebra, Exercise 2: EXERCISE - 1.2

Given $3x^2+x=1,$ find the value of $6x^3-x^2-3x+100$

Given $a^4+a^3+a^2+a+1=0$. Find the value of $a^{2000}+a^{2010}+1$.

Given that $f\left(x\right)$ is a polynomial of degree $3,$ and its remainders are $2x-5$ and $-3x+4$ when divided by $x^2-1$ and $x^2-4$, respectively. Find the $f(-3)$

Let $a$ and $b$ be two integers. Suppose $x^2-x-1$ is a factor of the polynomial $a{x}^5+b{x}^4+1$. Find the value of $a$.

Suppose $f$ is a function satisfying $f\left(x+x^{-1}\right)=x^6+x^{-6}$, for all $x\ne 0.$ Determine $400-f\left(3\right)$.

Suppose $x_1, x_2$ and $x_3$ are roots of ${\left(11-x\right)}^3+{\left(13-x\right)}^3={\left(24-2x\right)}^3$. What is the sum of $x_1+x_2+x_3?$

If $y+4=(x-2{)}^2, x+4=(y-2{)}^2$ and $x\ne y,$ then what is the value of $x^2+y^2$?

For how many integers is the number $x^4-51x^2+50$ negative?