Question

Find the remainder when 4x^3-12x^2+14x-3 is divided by 2x-1

Answer 1

According to remainder theorem, We know if f(x) is divided by ( x - a) , then remainder = f(a) ,
If f(x) is divided by (x-a) we have taken it (x -a) = 0 .So, Remainder would be f(a) .

Now, p(x) = 4x³ -12x²+14x - 3 .

If p(x) is divided by 2x-1 , then (2x-1) = 0 , x = 1/2 .So when p(x) is divided by (2x-1) , it leaves a remainder p(1/2)

p(1/2)
= 4(1/2)³ -12(1/2)²+14(1/2)-3
= 4(⅛)-12(1/4)+7-3
= 1/2 -3 + 7 - 3
= 1/2 +1
= 3/2

$\therefore$ The remainder when 4x³-12x²+14x-3 divided by 2x-1 is 3/2

Answer 2

Hi there !!

This can be done by two methods :

1. By long division method [ in the attachment ]

2. By using the remainder Theorem

We know,
2x - 1 = 0
2x = 1
x = 1/2

Substituting the values , we have ,

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

So,
we have ,

$\frac{1}{2} - 3 + 7 - 3$

$= \frac{1}{2} + 1 = \frac{3}{2}$
= 1.5

Thus,
the reaminder is 1.5