Division then Show Number of Real Roots

Use polynomial division and the discriminant or other argument to show that a polynomial equation has exactly one real root or no real roots.

5 questions · Standard +0.3

Sort by: Default | Easiest first | Hardest first

Find the quotient when $$x ^ { 4 } - 2 x ^ { 3 } + 8 x ^ { 2 } - 12 x + 13$$ is divided by $x ^ { 2 } + 6$ and show that the remainder is 1 .
Show that the equation $$x ^ { 4 } - 2 x ^ { 3 } + 8 x ^ { 2 } - 12 x + 12 = 0$$ has no real roots.

Find the quotient when $4 x ^ { 3 } + 8 x ^ { 2 } + 11 x + 9$ is divided by ( $2 x + 1$ ), and show that the remainder is 5 .
Show that the equation $4 x ^ { 3 } + 8 x ^ { 2 } + 11 x + 4 = 0$ has exactly one real root.

Find the quotient and remainder when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + 12 x + 6$$ is divided by ( $x ^ { 2 } - x + 4$ ).
It is given that, when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q$$ is divided by ( $x ^ { 2 } - x + 4$ ), the remainder is zero. Find the values of the constants $p$ and $q$.
When $p$ and $q$ have these values, show that there is exactly one real value of $x$ satisfying the equation $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q = 0$$ and state what that value is.

Find the quotient when $3 x ^ { 3 } + 5 x ^ { 2 } - 2 x - 1$ is divided by ( $x - 2$ ), and show that the remainder is 39 .
Hence show that the equation $3 x ^ { 3 } + 5 x ^ { 2 } - 2 x - 40 = 0$ has exactly one real root.

Find the quotient and remainder when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + 12 x + 6$$ is divided by ( $x ^ { 2 } - x + 4$ ).
It is given that, when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q$$ is divided by $\left( x ^ { 2 } - x + 4 \right)$, the remainder is zero. Find the values of the constants $p$ and $q$.
When $p$ and $q$ have these values, show that there is exactly one real value of $x$ satisfying the equation $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q = 0$$ and state what that value is.