Question

without actual division ,prove that 2x^4 - 8x^3 + 3x^2 + 2x - 9 is exactly divisible by x^2 - 4x +3

Answer 1

let f(x) = 2x^4 - 8x^3 + 3x^2 +12 -9
if f(x) is exactly divisible by x^2 -* 4x +3 = (x-1) (x-3) then f(1)=0 and f(3)=0
thus, 
f(1)= 2(1)^4 - 8(1)^3  3(1)^2 +12(1) -9
= 2 - 8 +3 +12 -9
=0
and, 
f(3)= 2(3)^4 -8(3)^3 + 3(3)^2 + 12(3) -9
=162 - 216 +27 + 36 - 9
=0
hence f(x) is exactly divisible by x^2 - 4x +3

Answer 2

Answer:

Step-by-step explanation:

let f(x) = 2x^4 - 8x^3 + 3x^2 +12 -9

if f(x) is exactly divisible by x^2 -* 4x +3 = (x-1) (x-3) then f(1)=0 and f(3)=0

thus, 

f(1)= 2(1)^4 - 8(1)^3  3(1)^2 +12(1) -9

= 2 - 8 +3 +12 -9

=0

and, 

f(3)= 2(3)^4 -8(3)^3 + 3(3)^2 + 12(3) -9

=162 - 216 +27 + 36 - 9

=0

hence f(x) is exactly divisible by x^2 - 4x +3

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