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 $x^4+k{x}^2+2x+9$, we get $1^4+k\ast 1^2+2\ast 1+9=7$. Solving this equation, we find that k=-5.

Step 3 ：Substitute x=-1 into the second polynomial $x^3+k{x}^2+2x+9$ to find the remainder when it is divided by x+1.

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

Step 5 ：Final Answer: The remainder when $x^3+k{x}^2+2x+9$ is divided by $x+1$ is $\boxed{{\displaystyle 1}}$.