Question

If the polynomial x⁴-6x³+16x²-25x+10 is divided by (x²-2x+k) the remainder comes out to be x+a, Find k & a ?

Answer 1

Solution

Given

• p(x) = x^4 - 6x³ + 16x² - 25x + 10

• g(x) = x² - 2x + k

• r(x) = x + a

Now we will find the remainder in terms of k and x and we shall equate the remainder to given r(x) = x + a.

( Refer to attachment )

Afer dividing p(x) by g(x) we get remainder r(x) :

→ Remainder r(x) = (-9 + 2k)x - [10 - 8k + k²]

→ Given, Remainder r(x) = x + a

From the above two equations we can equate them

⇒ (-9 + 2k)x - [10 - 8k + k²] = x + a

Comparing on both sides we get,

⇒ (-9 + 2k)x = x and ⇒ 10 - 8k + k² = a

Solving (-9 + 2k)x = x we get the value of 'k'

⇒ (-9 + 2k)x = x

⇒ - 9 + 2k = x/x

⇒ - 9 + 2k = 1

⇒ 2k = 1 + 9

⇒ 2k = 10

⇒ k = 10/2

⇒ k = 5

Substituting k = 5 in 10 - 8k + k² = a we get the value of 'a'

⇒ 10 - 8k + k² = a

⇒ 10 - 8(5) + (5)² = a

⇒ 10 - 40 + 25 = a

⇒ 35 - 40 = a

⇒ - 5 = a

⇒ a = - 5

Hence, the value of 'a' is - 5 and the value of 'k' is 5.

Answer 2

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