When the polynomial 2x3 kx2 5k 3 x 8 is divided by x 2 The remainder is 14 then the value of k is?

If the remainder when 2x3 + 3x2 – 5kx + 15 is divided by x – k is 2k3, then find the value of k.

1. ±√(15/2)
2. ±√(5/2)
3. ±√15
4. ±√2

Answer (Detailed Solution Below)

Option 1 : ±√(15/2)

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Given:

P(x) = Dividend = 2x3 + 3x2 – 5kx + 15

Divisor = x – k

Remainder = 2k3

Calculation:

As the remainder is 2k3 when divided by x – k

For x – k = 0 , x = k

P(k) = 2k3

2k3 + 3k2 – 5k2 + 15 = 2k3

⇒ –2k2 + 15 = 0

⇒ 2k2 = 15

⇒ k2 = 15/2

⇒ k = ±√(15/2)

∴ Value of k is ±√(15/2)

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Using remainder theorem, find the value of k if on dividing 2x3 + 3x2 - kx + 5 by x - 2, leaves a remainder 7

Let f(x) = 2x3 + 3x2 - kx + 5

Using remainder theorem,

f(2) = 7

`:.2(2)^3 + 3(2)^2 - k(2) + 5 = 7`

`:. 2(8) + 3(4) - k(2) + 5 = 7`

:. 16 + 12 - 2k + 5 = 7

∴ 2k = 16 + 12 + 5 - 7

∴ 2k = 26

∴ k = 13

f(x) = 2x2 + 3x2 – kx + 5
g(x) = x – 2, if x – 2 = 0, then x = 2
Dividing f(x) by g(x) the remainder will be
f(2) = 2(2)3 + 3(2)2 - k x 2 + 5
= 16 + 12 – 2k + 5
= 33 – 2k
Remainder = 7
∴ 33 – 2k = 7
⇒ 33 – 7 = 2k
⇒ 2k = 26
⇒ k = `(26)/(2)` = 13
∴ k = 13.

Concept: Remainder Theorem

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hello everyone the questions is that when the polynomial 2 x cube minus x square + 5 Y - 3 x minus 8 is divided by x minus 2 the remainder is 40 find the value of the polynomial is given to us first before before solving this we should know one concept that is if the polynomial f x is divisible bye x minus 1 then the four then we can say that remainder will be equal to that when x minus A is going to be equal to zero and whatever value of axle with the we have to put in the animal that is going to be reminder same cause we're going to apply here here the polynomial PX let se is given as 2 x cube minus x square

+ Y K minus 3 into x minus 8 which is divisible by x is equal to x minus 2 so from here first of all we have to take out the value of X equal to x minus 2 equal to zero and x is equal to 2 will check it to speak to you will replace all the expect to see it is going to be 2 into 2 cube that is 8 - ke into two square that is 4 + 5 ke minus 3 into 2 - 8 also this is the method to find the remainder is already given civil right here remainder value that is 14 is equal to 16 - 4K and here also will open the brackets 25 then 3 into 2 that is 6

- 8 so this is going to give us 16 - 6 that is 10 then 10 - 8 is 262 - 4K + 10 ke Sikke 14 so that we can clearly see from here that K6 ke is going to be equal to 12 which implies a value is 12 by 6 which is this is the value of k i hope you understand the solution thank you

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