The Factor Theorem

Show that the polynomial $5x^3-13x^2+17x-9$ has exactly one real root.

The polynomial $x^3+3x^2+4x+2$ is denoted by $f\left(x\right)$. Use the factor theorem to show that $x+1$ is a factor of $f\left(x\right)$.

The polynomial $4x^3+k{x}^2-65x+18$ is denoted by $f\left(x\right)$. Given that $x+2$ is a factor of $f\left(x\right)$. Write down the roots of $f\left(x^2\right)=0$

The polynomial $4x^3+k{x}^2-65x+18$ is denoted by $f\left(x\right)$. Given that $x+2$ is a factor of $f\left(x\right)$, solve the equation $f\left(x\right)=0$.

The polynomial $4x^3+k{x}^2-65x+18$ is denoted by $f\left(x\right)$. Given that $x+2$ is a factor of $f\left(x\right)$, find the value of $k$.

The polynomial $2x^3+a{x}^2+bx+6$ is denoted by $p\left(x\right)$.  Given that $(x-3)$ and $(x+2)$ are factors of $p\left(x\right)$, factorise $p\left(x\right)$ completely.

The polynomial $2x^3+a{x}^2+bx+6$ is denoted by $p\left(x\right)$.  Given that $(x-3)$ and $(x+2)$ are factors of $p\left(x\right)$, find the value of $a$ and value of $b$.

The polynomial $a{x}^3+b{x}^2-x+12$ is denoted by $p\left(x\right)$.  Given that $(x-3)$ and $(x+1)$ are factors of $p\left(x\right)$, find the other linear factor of $p\left(x\right)$.

The polynomial $a{x}^3+b{x}^2-x+12$ is denoted by $p\left(x\right)$.  Given that $(x-3)$ and $(x+1)$ are factors of $p\left(x\right)$, find the value of $a$ and value of $b$.

It is given that $\left(x+1\right)$ is a common factor of $\left(x^3+px+q\right)$ and $\left(x^3+\left(1-p\right)x^2+19x-2q\right)$.

Solve: $2x^4-11x^3+12x^2+x-4=0$

Solve: $x^4+2x^3-7x^2-8x+12=0$

Solve: $3x^3+17x^2+18x-8=0$

Solve: $2x^3-5x^2-13x+30=0$

Solve: $x^3+5x^2-17x-21=0$

Solve: $x^3-5x^2-4x+20=0$.

Given that $\left(x-1\right)$ and $\left(x+2\right)$ are factors of $x^4-x^3+p{x}^2-11x+q$. Factorise $x^4-x^3+p{x}^2-11x+q$ completely.

Given that $\left(x-1\right)$ and $\left(x+2\right)$ are factors of $\left(x^4-x^3+p{x}^2-11x+q\right)$. Find the value of $p$ and the value of $q$.

It is given that $\left(x+1\right)$ is a common factor of $\left(x^3+px+q\right)$ and $\left(x^3+\left(1-p\right)x^2+19x-2q\right)$. Find the value of $p$ and the value of $q$.

It is given that $\left(x+a\right)$ is a factor of $\left(x^3+4x^2+7ax+4a\right)$. Find the possible values of $a$.