How do you use the factor theorem to determine whether x-2 is a factor of $4x^3 – 3x^2 – 8x + 4$ ?

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Answer 1 · 2018-06-08T07:42:09

see below

the factor theorem states

$(x-a) " is a factor of "f(x) <=>f(a)=0$

$f(x)=4x^3-3x^2-8x+4$

$"we have " (x-2)=>a=2$

$f(2)=4xx2^3-3xx2^2-8xx2+4$

$f(2)=4xx8-3xx4-16+4$

$f(2)=32-12-16+4$

$f(2)=8!=0$

$:.x-2" is not a factor of "4x^3-3x^2-8x+4$

however by the remainder theorem when

$f(x)=4x^3-3x^2-8x+4" is divided by "(x-2) " the remainder is "8$

the factor theorem being a special case of the remainder theorem

Answer 2 · 2018-06-08T11:08:25

$"not a factor"$

$"if "x-2" is a factor then "f(2)=0$

$4(2)^3-3(2)^2-8(2)+4=8$

$"hence "x-2" is not a factor of "4x^3-3x^2-8x+4$

How do you use the factor theorem to determine whether x-2 is a factor of 4x^3 – 3x^2 – 8x + 44x^3 – 3x^2 – 8x + 4?