Using the remainder theorem, find the remainder when f(x) is divided by g(x) and verify the by actual division

Here, $f(x)=4 x^{3}-12 x^{2}+14 x-3$

g(x) = 2x - 1

from, the remainder theorem when f(x) is divided by g(x) = 2(x - 1/2), the remainder is equal to f(1/2)

Let, g(x) = 0

⟹ 2x - 1 = 0

⟹ 2x = 1

⟹ x = 1/2

Substitute the value of x in f(x)

$f(1 / 2)=4(1 / 2)^{3}-12(1 / 2)^{2}+14(1 / 2)-3$

= 4(1/8) - 12(1/4) + 4(1/2) - 3

= (1/2) - 3 + 7 - 3

= (1/2) + 1

Taking L.C.M

$=\left(\frac{2+1}{2}\right)$

= (3/2)

Therefore, the remainder is (3/2)