What is the coefficient of the second term in $ xyz+(x+1)(y)+4x$? The definition I found for a coefficient is 'the number multiplied by a variable.'
For the expression
$$
xyz + \left(x+1\right)\left(y\right) + 4x
$$
are both the factors $\left(x + 1\right)$ and $y$ considered variables?
In this case, I think the coefficient of the second term is $1$. Am I right $?$.
 A: Typically, when people order the terms (and refer to "the second term" or "the first term") they are ordering the terms from highest degree to lowest degree. To do this, you should expand the expression and look at the degrees of the terms. The variables here are $x$, $y$, and $z$. Note that the degree of the term is the sum of the exponents of the variables (in polynomials) or, in simpler words, the number of variables (or copies) being multiplied together. You expression expands to this:
$xyz+xy+y+4x$
Where the terms are $xyz$ with degree 3, $xy$ with degree 2, $y$ with degree 1, and $4x$ with degree 1. If we want the second highest degree, we would be talking about the term $xy$ which, as you said, has a coefficient of $1$.
A: I think a more appropriate definition for coefficient might be any multiplicative factor, as wikipedia suggests here. We usually use it to refer to the number that's multiplied to a variable in an expression. For example, in the expression $4x$, $4$ is typically referred to as a coefficient, and $x$ is a variable.
At the same time, variables can also be coefficients. For example, in the general form of a quadratic expression $ax^2 + bx + c$, we typically call $a$, $b$, and $c$ the coefficients, even though they are unknown. In some types of problems, they can actually be the unknown - for example, when fitting a parabola through a set of known points. Though, just as a caveat, it would be strange to refer to the independent variable ($x$ in this case) as a coefficient.
Also notice that in the quadratic above, we refer to $a$ as a coefficient, even though it's not multiplied to a variable - it's multiplied to $x^2$, another, slightly more complex expression.
My point is, a coefficient doesn't have to be limited to a number that's multiplied to a variable.
Going back to the problem at hand, in the expression $xyz+(x+1)(y)+4x$, there are three terms. The first is $xyz$, the second is $(x+1)(y)$, and the third is $4x$. In the same way that we could think of $x^2$ in the quadratic example above as a variable expression, and then think of $a$ as the coefficient on it, we can use the same type of logic here with respect to $xyz$ and $(x+1)(y)$. We can think of $xyz$ as a variable expression with coefficient $1$, and $(x+1)(y)$ as a variable expression with coefficient $1$. So, the coefficient on the second term is the $1$ that's the coefficient on $(x+1)(y)$.
