Question Type 3: Finding several parameter values given remainder and/or factors Exercises

For what value of $k$ is the quadratic $kx^2 - 7x + 2$ divisible by $x - 1$?

The cubic polynomial $P(x) = ax^3 - 2x^2 + cx + 18$ has a factor $(x - 2)$. When $P(x)$ is divided by $(x - 4)$, the remainder is 14. Find the values of $a$ and $c$.

Determine $k$ such that the cubic $kx^3 - x^2 + x - k$ is exactly divisible by $x - 1$.

Find $k$ if the remainder when $2x^2 + kx - 8$ is divided by $x - 3$ is 10.

Find $b$ and $c$ if the quadratic $x^2 + bx + c$ has factors $x + 1$ and $x - 2$.

Find $p$ and $q$ for the polynomial $px^3 + qx^2 + 5x - 2$ if it is divisible by $x - 1$ and leaves a remainder of 4 when divided by $x - 2$.

Determine $k$ such that $x^2 + kx - 12$ is exactly divisible by $x + 4$.

For the quartic $x^4 + ax^3 + bx^2 + 3x + 4$ the remainders when divided by $x - 1$ and $x + 1$ are 5 and 7 respectively. Find $a$ and $b$.

Find $a$ and $b$ if the quartic $x^4 - 2x^3 + ax^2 + bx + 5$ has factors $x - 1$ and $x + 2$.

Find the value of $k$ if the polynomial $kx^2 + 5x + 2$ leaves a remainder of 3 when divided by $x - 1$.

The cubic polynomial $x^3 + ax^2 + bx + 6$ has factors $x - 1$ and $x + 2$. Find $a$ and $b$.