R K Bansal Solutions for Exercise 1: EXERCISE

The polynomial$p{x}^3+4x^2-3x+q$ is completely divisible by ${\mathrm{x}}^2-1$; find the values of$\mathrm{p}$ and $\mathrm{q}$. Also, for these values of $\mathrm{p}$ and factorize the given polynomial completely.

Find the number which should be added to $x^2+x+3$ so that the resulting polynomial is completely divisible by $(\mathrm{x}+3)$.

When the polynomial $x^3+2x^2-5ax-7$ is divided by $(\mathrm{x}-1)$, the remainder is A and when the polynomial $x^3+a{x}^2-12x+16$ is divided by $(\mathrm{x}+2)$, the remainder is $\mathrm{B}$. the value of $\text{'}\mathrm{a}\text{'}$ if $2A+B=0.$

$(3\mathrm{x}+5)$ is a factor of the polynomial$(a-1)x^3+(a+1)x^2-(2a+l)x-15$. Find the value of$\text{'}\mathrm{a}\text{'}$ For this value of$\text{'}\mathrm{a}\text{'}$, factorise the given polynomial completely.

When divided by $\mathrm{x}-3$ the polynomials $x^3-p{x}^2+x+6$ and $2x^3-x^2-(p+3)x-6$ leave the same remainder. Find the value of $\text{'}\mathrm{p}\text{'}$.

Use the Remainder Theorem to factorise the following expression :

$2x^3+x^2-13x+6$

Using remainder theorem, find the value of $\mathrm{k}$ if on dividing $2{\mathrm{x}}^3+3{\mathrm{x}}^2-\mathrm{kx}+5$by $\mathrm{x}-2$, leaves a remainder $7$.

What must be subtracted from $16{\mathrm{x}}^3-8{\mathrm{x}}^2+4\mathrm{x}+7$ so that the resulting expression has $2\mathrm{x}+1$ as a factor ?