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 the values of k and a.
Question
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 the values of k and a.
Solution
To find the values of k and a, we can use the Remainder Theorem which states that the remainder of a polynomial f(x) divided by (x - c) is f(c).
Given that the remainder is (x + a), we can write it as (x - (-a)). This means that when the polynomial is divided by x2 – 2x + k, the remainder when x = 2 and x = -a is the same.
Let's find the value of a first:
- Set the polynomial equal to the remainder when x = -a:
(-a)^4 - 6(-a)^3 + 16(-a)^2 - 25(-a) + 10 = -a
Solving this equation gives a = 1.
Now, let's find the value of k:
- Set the polynomial equal to the remainder when x = 2:
2^4 - 62^3 + 162^2 - 25*2 + 10 = 2 + a
Solving this equation gives k = 5.
So, the values of a and k are 1 and 5, respectively.
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