Question

The polynomial $p{x}^3+4x^2-3x+q$ is completely divisible by $x^2-1$; find the value of p and q .Also for these values of p and q completely factories the given polynomial.

Answer 1

(1) let $f\left(x\right)=p{x}^3+4x^2-3x+q.$

Here $x^2-1=(x+1)(x-1)$ divides f(x)

$\therefore Atx=1,$

$f\left(1\right)=P(1{)}^3+4(1{)}^2-3\left(1\right)+q=0$

$\Rightarrow P+4-3+q=0.$

$\Rightarrow P+q+1=0$

$\Rightarrow$ P = -q-1 ______ (1)

and at x=-1

$f(-1)=P(-1{)}^3+4(-1{)}^2-3(-1)+q=0$

$\Rightarrow -p+4+3+q=0$

$\Rightarrow -(-q-1)+7+q=0$ [from (1)]

$\Rightarrow q+1+7+q=0.$

$\Rightarrow 2q=-8$

$\Rightarrow q=-4$

$\therefore$ from (1)

P =-(-4)-1 = 4-1=3

Hence, P = 3 , q = -4.