Question

If (x+1) and (x-1) are factors of ax³+ x²-2x+b , Find the value of a and b.

Answer 1

SOLUTION :-

p(x) = ax³+ x² - 2x + b

Given that,

( x + 1 ) and ( x - 1 ) are the factors of p(x) = ax³+ x² - 2x + b.

∴ p ( -1 ) = 0 , p ( 1 ) = 0

$\bullet\sf \ p(-1) = 0$

  $\longrightarrow\sf a\times (-1)^3+(-1)^2-2\times (-1)+b\\\\\longrightarrow -a+1+2+b= 0 \\\\\longrightarrow -a+b+3=0 \\\\\longrightarrow -a+b = - 3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ................(1)$

$\bullet\sf \ p(1) = 0$

   $\longrightarrow\sf a\times (1)^3+(1)^2-2\times1 + b = 0 \\\\\longrightarrow a+1-2+ b= 0 \\\\\longrightarrow a+b-1 = 0 \\\\\longrightarrow a+ b = 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ................(2)$

Add equation (1) and equation (2),

 $\longrightarrow\sf 2b = -3+1 \\\\\longrightarrow 2b = -2 \\\\\longrightarrow \bf b = -1$

Substitute the value of b in equation (2),

 $\longrightarrow\sf a+-1 = 1 \\\\\longrightarrow a-1 = 1 \\\\\longrightarrow\bf a = 2$

Value of a = 2

Value of b = -1