WHEN x^4+kx^2+2x+9 IS DIVIDED BY
x-1, THE REMAINDER IS 7. WHAT IS THE REMAINDER WHEN x^3+kx^2+2x+9 IS DIVIDED BY x+1

Step 1 ：The remainder theorem states that the remainder of a polynomial f(x) divided by (x-a) is f(a). So, we know that when we substitute x=1 into the first polynomial, the result is 7. We can use this to solve for k.

Step 2 ：Substitute x=1 into the equation \(x^4+kx^2+2x+9\), we get \(1^4+k*1^2+2*1+9=7\). Solving this equation, we find that k=-5.

Step 3 ：Substitute x=-1 into the second polynomial \(x^3+kx^2+2x+9\) to find the remainder when it is divided by x+1.

Step 4 ：Substitute x=-1 and k=-5 into the equation \(x^3+kx^2+2x+9\), we get \((-1)^3+(-5)*(-1)^2+2*(-1)+9=1\).

Step 5 ：Final Answer: The remainder when \(x^3+kx^2+2x+9\) is divided by \(x+1\) is \(\boxed{1}\).