Question

If the polynomial x^4 - 6x³ - 16x² + 25x + 10 is divided by another polynomial x² - 2x + k, the remainder comes out to be x + a, find k and a.​

Answer 1

Correct Question

If the polynomial f(x) = x⁴ - 6x³ + 16x² - 25x + 10 is divided by x² - 2x + k then remainder comes x + a. Find value of k and a.

Solution

If remainder is subtracted from dividend then divisor can divide the dividend.

→ x⁴ - 6x³ + 16x² - 25x + 10 - (x + a)

→ x⁴ - 6x³ + 16x² - 26x + 10 - a

(See the attachment)

Each getting 0,

→ x(2k - 10) = 0

→ 2k = 10 → k = 5

→ 10 - a - 8k + k² = 0

→ 10 - a - 40 + 25 = 0

→ - a - 5 = 0

→ a = - 5

Hence value of k is 5 and that of a is - 5.

Answer 2

It was given that the remainder is x+a.

The remainder we got is (-9+2x)x+(10-8k+k^2)

So when we compare the both remainders we get that..

x = (2k-9)x

x on L.H.S side and R.H.S side get cancel.

2k-9 = 1

2k = 1+9

2k = 10

k = 10/2

k = 5

Now, a = 10-8k+k^2

substitute the value of k in the equation above.

a = 10-8 (5)+25

a = 10-40+25

a = -30+25

a=-5

Therefore the values of k = 5 and a = -5.