Question 5:

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

Answer:

By division algorithm,
Dividend = Divisor × Quotient + Remainder
Dividend − Remainder = Divisor × Quotient

x⁴ – 6x³ + 16x² – 25x + 10 – x – a = x⁴ – 6x³ + 16x² – 26x + 10 – a will be perfectly divisible by x² – 2x + k.

Let us divide x⁴ – 6x³ + 16x² – 26x + 10 – a by x² – 2x + k

 

It can be observed that (–10 + 2k) x + (10 – a – 8k + k²) will be 0.

Therefore, (–10 + 2k) = 0 and (10 – a – 8k + k²) = 0

For (–10 + 2k) = 0,

2k = 10

And thus, k = 5

For (10 – a – 8k + k²) = 0

10 – a – 8 × 5 + 25 = 0

10 − a − 40 + 25 = 0
− 5 − a = 0
Therefore, a = −5
Hence, k = 5 and a = −5

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