Question

by remainder theorem
p(x) = 4x³ - 12x² + 5x - 4
and
g(x) = 2x - 1
p(x) is divided by g(x)​

Answer 1

To check if p(x) is divided by g(x)

First find zero of g (x)

2x - 1 = 0

→ x = 1/2

Now put this value of x in p (x). If p (x) becomes zero, then p(x) is divided by g(x) lest no.

p(1/2) = 4(1/2)³ - 12(1/2)² + 5(1/2) - 4

= 4/8 - 12/4 + 5/2 - 4

= 1/2 + 5/2 - 3 - 4

= 3 - 3 - 4

= -4

since p(x) ≠ 0 for x = 1/2, p(x) is not divisible by g(x)

Answer 2

Given :-

4x³ - 12x² + 5x - 4

To Find :-

Whether g(x) = 2x - 1 divisible by equation

Solution :-

g(x) = 2x - 1

0 = 2x - 1

0 + 1 = 2x

1 = 2x

1/2 = x

Now, Putting the value of x in p(x)

p(x) = 4x³ - 12x² + 5x - 4

0 = 4x³ - 12x² + 5x - 4

0 = 4(1/2)³ - 12(1/2)² + 5(1/2) - 4

0 = 4(1³/2³) - 12(1²/2²) + 5(1/2) - 4

0 = 4(1/8) - 12(1/4) + (5/2) - 4

0 = 4/8 - 12/4 + 5/2 - 4

0 = 4 - 24 + 20 - 32/8

0 = 24 + (-56)/8

0 = 24 - 56/8

0 = -32/8

0 ≠ -4

Therefore

It will be not divisible by 2x - 1