Factor Theorem

Write the degree of the following polynomial.
$13z^3-12z^2+11z+5$.

$(x-2)$ is a factor of expression $x^3+a{x}^2+bx+6$. When this expression is divided by $(x -3)$, it leaves the remainder $3$. Find the values of $a and b$.

Find $b and c$ such that, $\left(y+1\right) and \left(y+2\right)$ are the factors of the polynomial $y^3+b{y}^2-cy+10$.

Find the value of constants $a and b$, $\left(x-2\right) and \left(x+3\right)$ are both factors of the expression $x^3+a{x}^2+bx-12$.

What number must be added to  $2x^3-7x^2+ 2x$, so that the resulting polynomial leaves the remainder $-2$ when divided by $(2x -3)$?

The polynomials $3x^3-a{x}^2+5x-13$ and $(a+1)x^2-7x+5$ leaves the same remainder when divided by $(x -3)$. Find the value of $a$.

When divided by $(y-2)$, the polynomials  $p{y}^3+3y^2-13$ and $2y^3- 5y+p$ leaves the same remainder. Find the value of $p$.

When  $x^3-3x^2-kx+4$  is divided by $(x -3)$, then the remainder is $k$. Find the value of the constant $k$.

Using remainder theorem, find the remainder when  $f\left(x\right)=3x^4+2x^3-\frac{x^2}{3}-\frac{x}{9}+\frac{2}{27}$  is divided by $g\left(x\right) =\left(x+\frac{2}{3}\right)$.

Without actual division, find the remainder, if  $p\left(x\right) = 9x^2-6x+2$ is divided by $(3x-2)$.

Find the remainder (without division) on dividing  $2x^3+x^2- 7x+4$ by $(x -2)$.

What must be subtracted from $16x^3- 8x^2+4x+7$, so that the resulting expression has $(2x + 1)$ as a factor?

If $\left(x+2\right)$ and $\left(x+3\right)$ are the factors $x^3+ax+b$, find the value of $a+b$.

Find the value of the following polynomial at $x=-1$ .
$2x^2+5x+7$.

Find the zeroes of the following polynomial.
$7y-1$.

If $2y^3+p{y}^2+qy-6$ has a factor $\left(2y+1\right)$ and leaves the remainder $12$ when divided by $\left(y+2\right)$. Calculate the values of $p$ and $q$, hence factorize the expression completely.

If $\left(-p{x}^2+qx+6+x^3\right)$ has $\left(x-1\right)$ as a factor and leaves a remainder $4$ when divided by$\left(x+1\right)$, then find the values of $p$ and $q$.

Using factor theorem, factorize the polynomial $x^3+6x^2+11x+6$. Hence, solve the given polynomial equation

Factories $x^3-23x^2+142x-120$, using factor theorem.

Show that $\left(3z+10\right)$ is a factor of $9z^3-27z^2-100z+300$. Factorize the given polynomial completely by using factor theorem. $\left(3z+10\right)$ is a factor of it.