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[tex]\Large{\underline{\underline{\mathfrak{\bf{\green{Question}}}}}}[/tex]
If 4x⁴ - (a-1)x³ + ax² - 6x - 1 = 0 is divisible by 2x - 1 , find the value of " a " .
[tex]\Large{\underline{\underline{\mathfrak{\bf{\green{Solution}}}}}}[/tex]
If (2x - 1) is a factor of this polynomial ,
it means that value of x = 1/2 exits this equation .
So, Keep value of x = 1/2 in this equation , and calculate value of a
[tex]:\mapsto\sf{\:4.(\dfrac{1}{2})^4-(a-1).(\dfrac{1}{2})^3+a.(\dfrac{1}{2})^2-6.(\dfrac{1}{2})-1\:=\:0} \\ \\ \\ \\ :\mapsto\sf{\:4\times \dfrac{1}{16}\:-\:(a-1)\times \dfrac{1}{8}\:+\:a\times \dfrac{1}{4}-\:6\times \dfrac{1}{2}\:-\:1\:=\:0} \\ \\ \\ \\ :\mapsto\sf{\:\dfrac{1}{4}\:+\:\dfrac{1}{8}\:-\:\dfrac{6}{2}\:-\dfrac{a}{8}\:+\:\dfrac{a}{4}\:=\:0} \\ \\ \\ \\ :\mapsto\sf{\:\dfrac{(2+1-24)}{8}\:+\:\dfrac{(-a+2a)}{8}\:=\:0} \\ \\ \\ \\ :\mapsto\sf{\:\dfrac{a}{8}\:=\:-\dfrac{-21}{8}} \\ \\ \\ \\ :\mapsto\sf{\:a\:=\:\dfrac{-21}{8}\times 8} \\ \\ \\ \\ :\mapsto\sf{\pink{\:a\:=\:-21}}[/tex]
[tex]\Large{\underline{\mathfrak{\bf{\orange{Thus}}}}}[/tex]