If $x+7$ is a factor of $cx^3 + 19x^2 - 3cx + 35$, find the constant $c$.
I notice that this is a polynomial of degree 3, and one of its factors is given as $x+7$. That means that $x=-7$ is a root of the polynomial, or in other words, plugging in $x=-7$ should make the polynomial zero. So I will use that idea to find $c$. I substitute $x=-7$ into the polynomial and set it equal to zero: $c(-7)^3 + 19(-7)^2 - 3c(-7) + 35 = 0$. I simplify the powers and the signs: $-343c + 931 + 21c + 35 = 0$. I combine like terms: $-322c + 966 = 0$. I isolate $c$ by subtracting 966 from both sides: $-322c = -966$. I divide both sides by -322 to get $c$: $c = 3$. This is the value of $c$ that makes $x+7$ a factor of the polynomial. 