If the polynomial x⁴ - 6x³ + 16x² - 25x + 10 is divided by another polynomial x² - 2x + k, the remainder comes out to be x + a, find k and a.

Solution:

The given polynomial is p(x) = x⁴ - 6x³ + 16x² - 25x + 10

We can solve this by using division algorithm i.e, Dividend = Divisor × Quotient + Remainder

Let us divide and equate the obtained remainder with (x + a)

Now, it is given that p(x) when divided by x² – 2x + k  leaves (x + a) as remainder.

Let us equate the remainder with x + a (as given in the question)

(-9 + 2k)x + 10 - 8k + k² = x + a

Let us compare the coefficient of both LHS and RHS.

-9 + 2k = 1

⇒ 2k = 10 

⇒ k = 5 ----- (1)

Also, 10 - 8k + k² = a ----- (2)

As we obtained the value of k, let us substitute in equation (2) to find the value of a.

a = 10 - 40 + 25

a = - 5

Therefore, the value of k is 5 and a is - 5.

☛ Check: NCERT Solutions Class 10 Maths Chapter 2

Video Solution:

If the polynomial x⁴ - 6x³ + 16x² - 25x + 10 is divided by another polynomial x² - 2x + k, the remainder comes out to be x + a, find k and a.

NCERT Solutions Class 10 Maths Chapter 2 Exercise 2.4 Question 5

Summary:

If the polynomial x⁴ - 6x³ + 16x² - 25x + 10 is divided by another polynomial x² - 2x + k, the remainder comes out to be x + a. The values of k and a are 5 and - 5 respectively.

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