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.
Given,
Divisor = x2 – 2x + k
Dividend = x4 – 6x3 + 16x2 – 25x + 10
Remainder = x + a
As we know that,
Dividend = divisor quotient + remainder
x4 – 6x3 + 16x2 – 25x + 10 = x2 – 2x + k quotient + (x + a)
x4 – 6x3 + 16x2 – 25x + 10 – (x + a) = x2 – 2x + k quotient
x4 – 6x3 + 16x2 – 26x + 10 – a = quotient
x2 – 2x + k
If the polynomial x4 – 6x3 + 16x2 – 26x + 10 – a is divided by x2 – 2x + k remainder comes out to be zero.
Therefore, By equating the remainder with zero, we have
(-10 + 2k) = 0 => 2k = 10 => k = 5
Or, 10 – a – 8k + k2 = 0
Putting the value of k, we get
10 – a – 8(5) + (5)2 = 0
10 – a – 40 + 25 = 0
- a – 5 = 0 => a = -5
Hence, k = 5 and a = -5
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