Answer to Question #93109 in Algebra for Unknown287159

Question

Find the values of p and q if (x-1) and (x+2) are both factors px*3 +4x*2+qx -6

Answer 1

Using the factor theorem, if $f(x)=px^3+4x^2+qx-6$ has factors $(x-1)$ and $(x+2)$, then

$f(1)=0, f(-2)=0$.

$\begin{cases} p\cdot 1^3 +4\cdot 1^2 +q\cdot 1-6=0 \\ p\cdot (-2)^3 +4\cdot (-2)^2 +q\cdot (-2)-6=0 \end{cases}$

$\begin{cases} p+4 +q-6=0 \\ -8p +16 -2q-6=0 \end{cases}$

$\begin{cases} q=2-p \\ -8p -2q=-10 \end{cases}$

$\begin{cases} q=2-p \\ 4p+q=5 \end{cases}$

$\begin{cases} q=2-p \\ 4p+2-p=5 \end{cases}$

$\begin{cases} q=2-p \\ 3p=3 \end{cases}$

$\begin{cases} p=1 \\ q=1 \end{cases}$

Answer: $p=1,q=1$