Question

What must be subtracted from 2x⁴-5x³+5x+40 so that the result is divisible by 2x-1.

Answer 1

Since 2x2 + 3x - 2 is of degree 2, so when

p(x) = 4x4 - 2x3 - 6x2 + x - 5 is divided by

q(x) = 2x2 + 3x - 2, the remainder should be a linear expression (degree of remainder < degree of divisor).

Let the remainder r(x) = ax + b for exact division this remainder should be subtracted from p(x) Now let

f(x) = p(x) - r(x)

= 4x4 - 2x3 - 6x2 + x - 5 - (ax + b)

f(x) = 4x4 - 2x3 - 6x2 + (1 - a) x - 5 - b

Again, we haveq(x) = 2x2 + 3x - 2

= 2x2 + 4x - x - 2

= 2x (x + 2) - 1(x + 2)

= (x + 2) (2x - 1)

Since x + 2 and 2x - 1 are factors of q(x) and f(x) is exactly divisible by q(x), hence (x + 2) and (2x - 1) are also factors of f(x), i.e.

f(-2) = 0 and f(

) = 0

... f(-2)=4(- 2)4 - 2(- 2)3 - 6(- 2)2 + (1 - a) (- 2) - b - 5 = 0

⇒ 64 + 16 - 24 - 2 + 2a - b - 5 = 0

⇒ 2a - b = - 49 …… (i)

Again f(

) = 0

⇒ 4 (

)4- 2(

)3 - 6 (

)2 + (1 - a) x

- b - 5 = 0

⇒ -

-

+

-

-b - 5 = 0

⇒ -

- b - 6 = 0

⇒ a + 2b = - 12 …… (ii)

Solving eqns (i) and (ii), we get a = - 22 and b = 5.

... r(x) = - 22x + 5 should be subtracted.

Step-by-step explanation:

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Answer 2

Answer:

U should Subtract

$\frac{346}{8}$ =$\frac{173}{4}$

As it is left as remainder.