Question Type 2: For small polynomials, given a remainder or factor, finding a parameter's value Exercises

Determine $r$ such that $x^4 + r x^2 - x +4$ has remainder $0$ when divided by $x - 1$.

The polynomial $4x^3 + ax^2 - 7x + 1$ has a factor $x - 1$. Find the value of $a$.

Given $g(x)=6x^4 + dx^3 - x +5$ leaves a remainder of $7$ when divided by $(x-1)$, find the value of $d$.

The polynomial $Q(x)=x^4 -4x^2 + kx +5$ gives a remainder of $13$ when divided by $(x-2)$. Find the value of $k$.

Determine the value of $b$ given that $P(x)=2x^4 -3x^2 + bx -6$ leaves a remainder of $5$ when divided by $(x-2)$.

Given that $P(x) = x^3 + ax^2 - x + 2$ leaves a remainder of 5 when divided by $(x-1)$, find the value of $a$.

Find $c$ if $f(x) = x^3 + cx^2 - 4x + 3$ is exactly divisible by $x - 3$. [3 marks]

If $5x^3 - mx^2 + 2x + 1$ gives a remainder of $-3$ when $x = -1$, find the value of $m$.

Find $k$ if the polynomial $2x^3 + kx^2 -5x +1$ is exactly divisible by $x - 1$.

Find $k$ such that $x^5 + kx^3 - x + 2$ is divisible by $x + 1$.

Find the value of $b$ if the polynomial $3x^2 + bx + 10$ leaves a remainder of $3$ when divided by $(x - 2)$.

The polynomial $h(x) = 3x^3 - px + 4$ has a factor $x + 2$. Determine the value of $p$.