Question

if (x+2) is a factor of x^3-2ax^2+ax -2​

Answer 1

$\large{\underline{\underline{\red{\sf{Given: }}}}}$

• $\tt{x+2\:is\:a\:factor\:of\:x^3-2ax^2+ax-2.}$

$\large{\underline{\underline{\red{\sf{To\:Find: }}}}}$

• $\tt{The\:value\:of\:a .}$

$\large{\underline{\underline{\red{\sf{ Answer:}}}}}$

$\tt{Given\:that\:(x+2)\:is\:a\:factor\:of\:a\: polynomial.}$$\tt{So\:on\:putting\:x=(-2)\:whole\: polynomial}$$\tt{will\:become\: zero.}$

$\purple{\tt{\leadsto Here\:are\: the\:steps:}}$

$\tt{\orange{Step\:1\:\::-\:\green{Equate\:the\:factor\:with\:zero. }}}$

$\tt{\orange{Step\:2\:\::-\:\green{ Put\:that\:value\:in\:the\: polynomial.}}}$

$\tt{\orange{Step\:3\:\::-\:\green{ Equate\:the\: polynomial\:with\:0.}}}$

$\tt{\orange{Step\:4\:\::-\:\green{ Solve\:linear\:equ^n\:to\:find\:a.}}}$

__________________________________________

$\underline{\pink{\sf{\mapsto Following\:the\:above\:steps:-}}}$

$\tt{\implies x+2=0.}$

$\tt{\implies x =0-2.}$

$\underline{\boxed{\red{\bf{\dag \:\:\:\:x \:\:=\:\: (-2)\:\:\:\:}}}}$

$\underline{\blue{\sf{\mapsto Putting\:this\:value\:in\:the\:given\: polynomial:-}}}$

$\tt{\implies x^3-2ax^2+ax-2=p(x) \:\;[say].}$

$\tt{\implies (-2)^3-2\times(-2)^2\times a+a\times(-2)-2=0 .}$

$\tt{\implies -8-2\times4a-2a-2=0.}$

$\tt{\implies -10-8a-2a=0.}$

$\tt{\implies-10a-10=0 .}$

$\tt{\implies -10(a+1)=0.}$

$\tt{\implies (a+1)=\dfrac{0}{-10}.}$

$\tt{\implies a+1=0.}$

$\tt{\implies a=0-1 .}$

$\underline{\boxed{\red{\bf{\large{\dag} \:\:\:\:a \:\:=\:\: (-1)\:\:\:\:}}}}$

$\underset{\tt{\orange{Required\:Answer}}}{\pink{\underbrace{\underline{\underline{\blue{\sf{Hence\:value\:of\:a\:is\:1. }}}}}}}$