Question

If on dividing the polynomial 2x3 + kx2 – (5x – 3)x +8 by x + 2, the remainder is 30, then the value of k is:
(a) 8
(b) 9
(c) 10
(d) 11

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Answer 1

Answer:

On dividing the polynomial 2x³+kx²-(5x-3)x+8 by x + 2, the remainder is 30, then the value of k is (a) 8.

Step-by-step explanation:

Let f(x) = 2x³+kx²-(5x-3)x+8       (i)

From the question when f(x) is divided by (x+2) the remainder is 30.

Now, using Remainder Theorem which states that when a polynomial f(x) is divided by a factor (x-a) which may or may not be and element of the polynomial, a smaller polynomial along with a remainder is obtained. This remainder obtained is a value of f(x) at x=a i.e. f(a).

Hence, here x-a = x+2

or, a+2 = 0

or, a = -2

∴ f(a) = f(-2)

From equation (i) we get,

f(-2) = 2×(-2)³ + k×(-2)² - [{5×(-2) - 3} × -2] + 8

or, f(-2) = 2×(-2×-2×-2) + k×(-2×-2) - [{-10 - 3} × -2] + 8

or, f(-2) = 2×-8 + k×4 - [-2×-13] + 8

or, f(-2) = -16 + 4k - [26] + 8

or, f(-2) = 4k - 2

Now, According to the problem,

4k - 2 = 30

or, 4k = 32

or, k = 32/4

or, k = 8

Hence, on dividing the polynomial 2x³+kx²-(5x-3)x+8 by x + 2, the remainder is 30, then the value of k is 8.