Question

If the polynomial f (X)=x^4-6x^3+16x^2-25x+10 is divided by another polynomial x^2-2x+k, the remainder comes out to be X+a, find k and a

Answer 1

Step-by-step explanation:

Given If the polynomial f (X)=x^4-6x^3+16x^2-25x+10 is divided by another polynomial x^2-2x+k, the remainder comes out to be X+a, find k and a

• We have f(x) = x^4 – 6x^3 + 16x^2 – 25x + 10
•  Also divisor = x^2 – 2x + k
• Remainder = x + a
• Subtracting remainder from f(x) we get
• x^4 – 6x^3 + 16x^2 – 25 x + 10 – (x + a)
• x^4 – 6x^3 + 16x^2 – 26x + 10 – a = f(y)
• now f(y) is divisible by divisor x^2 – 2x + k
• So we get
•                                           x^2 – 4x + (8 – k)
• ----------------------------------------------------------------------------------------
•    x^2 – 2k + k) x^4 – 6x^3 + 16x^2 – 26x + 10 – a
•                            x^4 – 2x^3 + kx^2
•                     --------------------------------------------------------------------------
•                                    -4x^3 + (16 + k)x^2 – 26 x + 10 – a
•                                     -4x^3 + 8x^2           - 4kx
•                           -------------------------------------------------------------------------
•                                               (8 – k)x^2 – (26 – 4k)x + 10 – a
•                                                (8 – k)x^2 – (16 – 2k)x + (8k – k^2)
•                                    ----------------------------------------------------------------------
•                                                                (- 10 + 2k)x + (10 – a – 8k + k^2)
• Now remainder is zero
•              -(10 + 2k)x + (10 – a – 8k + k^2) = 0  
• Now – 10 + 2k = 0 and 10 – a – 8k + k^2 = 0
•        So k = 5 and 10 – a – 8(5) + 5^2 = 0
•                           10 – a – 40 + 25 = 0
•                             Or a = - 5

Therefore k = 5 and a = - 5

Reference link will be

https://brainly.in/question/2159029