Manipur Board Solutions for Chapter: Polynomials, Exercise 1: EXERCISE 2.1

Divide $p\left(x\right)$ by $d\left(x\right)$ and then verify division algorithm.

$p\left(x\right)=x^4-5x^2+6 \text{and} d\left(x\right)=2-x^2$

Check whether the first polynomial is a factor of the second polynomial by actual division.

$x^2-x+1, x^3+1$

Check whether the first polynomial is a factor of the second polynomial by actual division.

$x^2-3, 2x^4+3x^3-2x^2-9x-5$

Check whether the first polynomial is a factor of the second polynomial by actual division.

$t^2+3t+1, 3t^4+5t^3+2+2t-7t^2$

Check whether the first polynomial is a factor of the second polynomial by actual division.

$1+y^3+3y, -1-4y^3+y^5+y^3+3y$

Check whether the first polynomial is a factor of the second polynomial by actual division.

$7+3x, 3x^3+7x$

On dividing $x^3-3x^2+x+2$ by a polynomial $f\left(x\right)$, the quotient and remainder are $x-2$ and $4-2x$ respectively. Find $f\left(x\right)$.

When a polynomial $p\left(x\right)$ is divided by $3x-1$, the quotient and remainder are $x^2+2x-3$ and $5$ respectively. Find $p\left(x\right)$.