Question

if X + 1 and x minus 1 are the factor of p x cube + x square + 2 X + Q then the find value of P and Q​

Answer 1

$\huge{\mathfrak{Question:-}}$

If x + 1 and x - 1 are the factor of px³ + x² + 2x + Q then the find value of P and Q.

$\huge{\mathfrak{Solution:-}}$

​$\mathbf{Given:-}$

x - 1 and x + 1 are the factors of px³ + x² + 2x + q.

​$\mathbf{To\;find:-}$

Value of p and q.

$\sf{So,}$

$\sf{x + 1=0}$

$\sf{x=-1}$

$\sf{p(-1)=p\times (-1)^{3}+(-1)^{2}+2(-1)+q}$

$\sf{\implies -p+1-2+q}$

$\sf{\implies -p-1+q\;\;\;\;\;\;....(1)}$

$\sf{Then,}$

$\sf{x-1=0}$

$\sf{\implies x=1}$

$\sf{p(1)=p\times (1)^{3}+(1)^{2}+2\times 1+q}$

$\sf{\implies p+1+2+q}$

$\sf{\implies p+3+q\;\;\;\;\;\;...(2)}$

$\sf{Adding\;both\;the\;equation\;(1)\;and\;(2)\;we\;get,}$

$\sf{\implies -p-1+q+p+3+q=0}$

$\sf{\implies 2+2q=0}$

$\sf{\implies 2q=-2}$

$\sf{\implies q=-1}$

$\sf{Then\;put\;the\;value\;of\;q\;in\;equation\;(2)\;we\;get,}$

$\sf{p+3+q=0}$

$\sf{\implies p+3-1=0}$

$\sf{\implies p+2=0}$

$\sf{\implies p = -2}$

$\sf{So,}$

$\boxed{\sf{q=-1}}$

$\boxed{\sf{p=-2}}$

Answer 2

Solution:-

Given:-

( x + 1) and ( x - 1) are the Factor of (Px³ + x² + 2x + Q).

To Find :-

Value of P and Q.

Find:-

Since, ( x + 1) is the Factor.

=) ( x + 1) = 0

=) x = -1.

Now,

=) P (x) = Px³ + x² + 2x + Q

=) 0 = P (-1)³ + (-1)² + 2(-1) + Q.

=) 0 = -P +1 - 2 + Q

=) 0 = Q - P - 1.

=) 0 = Q - P - 1

=) Q - P = 1

=) Q = P +1. ----------(1).

Now, ( x -1) is also the Factor.

=) x - 1 = 0

=) x = 1.

P (1) = P(1)³ + (1)² + 2(1) + Q

=) 0 = P + 1 + 2 + Q

=) P + Q = -3

Putting ( Q = P+1). we get,

=) P + P + 1 = -3

=) 2P = -4

=) P = -2.

Putting ( P = -2) in Eq (1). we get,

=) Q = P + 1

=) Q = -2 + 1

=) Q = -1.

Hence,

P = -2

and Q = -1.