Question

f(x) = x³ - 6x² + 2x - 4, g(x) = 1- 2x find its reminder with the remainder theorem and verify the answer with long division ​

Answer 1

Answer:

Explanation:

Given polynomial,

⇒ f(x) = x³ - 6x² + 2x - 4

Which is to be divided by g(x) = 1 - 2x

By remainder theorem,

⇒ g(x) = 1 - 2x

⇒ 1 - 2x = 0

⇒ 1 = 2x

⇒ ½ = x

On substituting the value of ‘x’ in p(x) :-

⇒ p(½) = (½)³ - 6(½)² + 2(½) - 4

⇒ ⅛ - 6(¼) + 1 - 4

⇒ ⅛ - 3/2 - 3

⇒ -35/8

Check the attached file for verification by actual division.

Answer 2

Answer:

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Step-by-step explanation:

Solution :-

1 - 2x = 0

- 2x = - 1

2x = 1

x = 1/2

p(x) = x³ - 6x² + 2x - 4

p(1/2) = (1/2)³ - 6*(1/2)² + 2*(1/2) - 4

= 1/8 - 6*(1/4) + 1 - 4

= 1/8 - 6/4 + 1 - 4

= 1/8 - 3/2 + 1 - 4

Taking L.C.M. of the denominators and then solving it.

= (1 - 12 + 8 - 32)/8

= - 35/8

Remainder is - 35/8

Now, actual division.

- 6x² - 1

_________________

1 - 2x ) x³ - 6x² + 2x - 4 (

12x³ - 6x²

- +

______________

- 11x³ + 2x - 4

+ 2x - 1

- +

_______________

- 11x³ - 3

_______________

Remainder is - 11x³ - 3

Verification :

⇒ - 11x³ - 3

⇒ - 11*(1/2)³ - 3

⇒ - 11*1/8 - 3

⇒ - 11/8 - 3/1

⇒ Taking L.C.M. of the denominator and then solving it.

⇒ (- 11 - 24)/8

⇒ - 35/8

Verified.