M L Aggarwal Solutions for Chapter: Polynomials, Exercise 7: Chapter Test

Form quadratic polynomials whose one zero is $8$ and the product of zeroes is $56$.

If $\alpha$ and $\beta$ are zeroes of the polynomial $3x^2-4x-7$, then form a quadratic polynomial whose zeroes are $\frac{1}{\alpha }$ and $\frac{1}{\beta }$.

Divide $\left(6+19x+x^2-6x^3\right)$ by $\left(2+5x-3x^2\right)$ and verify the division algorithm.

On dividing the polynomial $3y^3-4y^2-3y+25$ by a polynomial $g\left(y\right)$, the quotient and remainder were $3y+5$ and $5$, respectively. Find $g\left(y\right)$.

Find the polynomial of least degree which should be subtracted from the polynomial $x^4+2x^3-4x^2+6x-3$ so that it is exactly divisible by $x^2-x+1$.

If $x^3+x^2-2x-3=(x-2)\left(x^2+ax+b\right)+5$, then find the values of $a$ and $b$ (using the concept of equality of polynomials).

Verify that $-1$ is a zero of the polynomial $p\left(x\right)=x^3-2x^2-x+2$. Obtain all the zeroes of the polynomial $p\left(x\right)$.

If two zeroes of the polynomial $x^4-6x^3-26x^2+138x-35$ are $2\pm \sqrt{3}$, find the other zeroes.