If the polynomial, 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 + a, please work the complete solution instead of giving simply an answer.
To find the value of k + a, we need to divide the given polynomial, x^4 - 6x^3 + 16x^2 - 25x + 10, by the polynomial x^2 - 2x + k and find the remainder as x + a.
First, let's perform polynomial long division:
x^2 - 2x + k | x^4 - 6x^3 + 16x^2 - 25x + 10
- (x^4 - 2x^3 + kx^2)
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-4x^3 + (16-k)x^2 - 25x
+ (4x^3 - 8x^2 + 4kx)
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(24 - k)x^2 - 25x + 10
- ((24 - 48 + 24k - k^2)x + (48 - 25k))
The remainder is given by (24 - k)x^2 - 25x + 10 - ((24 - 48 + 24k - k^2)x + (48 - 25k)). For this remainder to be x + a, the corresponding coefficients need to be equal.
Equating the corresponding coefficients:
24 - k = 0 ---> k = 24
-25 = 1 ---> a = -26
Therefore, k + a = 24 - 26 = -2.
Hence, the value of k + a is -2.
by using long division,
divide x^4-6x^3+16x^2-25x+10with x^2-2x+k,
then you will find:
remainder=(-9+2k)x +10-8k+k^2
comparing with x + a,
-9+2k=1,k=5
a=10-8k+k^2,a=-5
therefore k + a=5-5=0