M L Aggarwal Solutions for Chapter: Polynomials, Exercise 4: Exercise 2.4

Show that $x-3$ is a factor of the polynomial $x^3+x^2-17x+15$.

Without actual division, show that $2x^3+13x^2+x-70$ is exactly divisible by $x-2$.

Use factor theorem to determine whether $g\left(x\right)=x+1$ is a factor of $p\left(x\right)=2x^3+x^2-2x-1$ or not.

Use factor theorem to determine whether $g\left(x\right)=x-3$ is a factor of $p\left(x\right)=x^3-4x^2+x+6$ or not.

Using factor theorem, find the value of $a$ if the polynomial $p\left(x\right)=2x^4-a{x}^3+4x^2-x+2$ is divisible by $\left(2x+1\right)$.

By actual division, show that $\left(x+2\right)$ is a factor of the polynomial $p\left(x\right)=2x^4+x^3-14x^2-19x-6$. Verify it using factor theorem.

If $\left(x-1\right)$ and $\left(x+1\right)$ are factors of $a{x}^3+x^2-2x+b$, then find the values of $a$ and $b$.

If $2x^3+a{x}^2+bx-6$ has $\left(x-1\right)$ as a factor and leaves a remainder $2$ when divided by $\left(x-2\right)$, then find the values of $a$ and $b$.