In a polynomial, if $p(x)$ is divisible by $x$, why is the constant $0$? Here's the problem.
$p(x) = 3(x^2 + 10x + 5) − 5(x − k)$
In the polynomial $p(x)$ defined above, $k$ is a constant. If $p(x)$ is divisible by $x$, what is the value of $k$?
In the answer it explained that because "$p(x)$ is divisible by $x$" we know that the constant is $0$. But why is that true and what is the constant?
 A: That means $P(x) = xQ(x) \to P(0) = 0 \to k = -3$
A: If $p(x)$ is divisible by $x$, $p(x) = q x$ where q is another polynomial in $x$. In particular, if $x=0$, $p(0) = q 0 = 0$. Then $p(0) = 15 - 5(-k) = 15 + k$. Now you know that $ 15 + k =0 $. So $k = -3.$
A: Your function is a quadratic and a quadratic  can only have linear factors, one which you specified is $x$, the other we will call $L(x)$
$$xL(x)=3x^2-25x+5k+15$$
Set $x=0$ to get:
$$0=5k+15$$
Another way you can think about it:
When you divide your polynomial by $x$ you get:
$$3x-25+\frac{5k+15}{x}$$
The numerator of  last term is a remainder and we need to get rid of it by setting it equal to zero.
$$5k+15=0$$
$$k=-3$$
A: When it says $p(x)$ is divisible by $x$ it means everything in the equation must go away, otherwise saying everything must have an $x$ in the term. In the expanded which is $p(x) = 3 x^{2} + 25 x + 15 + 5 k$, $15$ and $5 k$ need to go away because they are NOT divisible by $x$. So you set $15 + 5 k=0$ and solve. You should end up with $k = -3$ and that's your answer!
A: Very helpful explanations, here is my twist on describing the logic...
If something can be nicely divisible by a quantity then that means the remainder must be zero.  Example: 21 can be nicely divisible by 3, 21/3 = 7 with a remainder of 0 ("7 R 0", or 7 + 0/3).  Contrary, 22 cannot be nicely divisible by 3, 22/3 does not have a remainder of zero, 22/3 = "7 R 1", or 7 + 1/3.
Go back to the problem and use this reasoning as a template for the solution.
Since we know that p(x) is divisible by x (nicely), then we know that the quantity that remains in terms of x must be equal to zero.  Use the expanded version of p(x) then divide it by x.  Notice that only one component of this result is left with "x" in the denominator, (15+5k)/x .  In other words, because (15+5k)/x is the only component with "x" in the denominator, it is not "nice" and is the remainder.
The problem tell us that the remainder must be "0" (because p(x) is divisible by x).
So, the simple equation, (15+5k)/x, must be equal to 0.  Now just solve for k.
I hope this long explanation helps... it helped with my son's understanding.
