find the value of k so that the remainder is 5. X^2+8X+17 divided by X+K
f(-k) = k^2 - 8k + 17
you want f(-k) = 5
k^2 - 8k + 12 = 0
(k-6)(k-2) = 0
so, for k = 2 or 6,
(x^2 + 8x + 17)/(x-2) = x+6 R 5
(x^2 + 8x + 17)/(x-6) = x+2 R 5
oops. we divide by x+k, so k = -2 or -6
X^2+8X+17
To find the value of k such that the remainder is 5 when dividing \(x^2 + 8x + 17\) by \(x + k\), we can use the Remainder Theorem.
The Remainder Theorem states that if you divide a polynomial \(f(x)\) by \(x - a\), the remainder is \(f(a)\).
Given that we want the remainder to be 5, we need to find a value of \(k\) such that:
\((x^2 + 8x + 17) \mod (x + k) = 5\)
To solve this, we substitute \(x = -k\) into the polynomial:
\((-k)^2 + 8(-k) + 17 = 5\)
Simplifying, we have:
\(k^2 - 8k + 17 = 5\)
Rearranging the equation to isolate \(k\), we have:
\(k^2 - 8k + 12 = 0\)
Now we can solve this quadratic equation by factoring or using the quadratic formula. Factoring gives us:
\((k - 2)(k - 6) = 0\)
So the solutions are: \(k = 2\) or \(k = 6\).
Therefore, the possible values of k so that the remainder is 5 are \(k = 2\) and \(k = 6\).