Multiplying a polynomial by a monomial is a direct application of the distributive and associative properties. Recall that the distributive property says that

$a(b + c) = ab + ac$

for all real numbers $a,b$ and $c.$ The associative property says that

$(ab)c=a(bc)$

for all real numbers $a,b$ and $c.$ 

As we will treat variables in the same way as real numbers, the same properties hold whenever $a,b$ and/or $c$ is a variable. So for the multiplication of a monomial with a polynomial we get the following procedure:

Multiply every term of the polynomial by the monomial and then add the resulting products together. 

For example,

$\begin{aligned} 3x^2(4x-\pi)&=3x^2 \cdot 4x-3x^2 \cdot \pi \\&=12x^3-3\pi x^2. \end{aligned}$

To multiply a polynomial $P(x) = M_1(x) + M_2(x) + \ldots + M_n(x)$ with a polynomial $Q(x) = N_1(x) + N_2(x) + \ldots + N_k(x)$, where both are written as a sum of monomials of distinct degrees, we get

$\begin{aligned} P(x)Q(x) &= (M_1(x)+M_2(x)+ \ldots + M_n(x))Q(x) \\ &=M_1(x)Q(x) + M_2(x)Q(x) + \ldots + M_n(x)Q(x) \\ &=M_1(x)N_1(x) + \ldots + M_1(x)N_k(x) + \ldots \\ &+M_n(x)N_1(x) + \ldots + M_n(x)N_k(x) \end{aligned}$

and we see that this equals the sum of the products of the terms, where every term of $P(x)$ is multiplied exactly once with every term of $Q(x)$. Notice that since the highest degree term of $P(x)$ is multiplied with the highest degree term of $Q(x)$ we have that the degree of the product equals the sum of the degrees, since

$a^na^m=a^{m+n}$

for all real numbers (and variables) $a$ and all non-negative integers $m$ and $n$.

For convenience, we will use the commutative property of addition to write the expression so that we start with the terms containing $M_1(x)$ and end with the terms containing $M_n(x)$.

This method is commonly called the FOIL method, where we multiply the First, Outside, Inside, and Last pairs in the expression, and then add the products of like terms together.

For example, to find the product of $(2x+3)(x−4)$, use FOIL and then add the products together: 

$\begin {aligned} 2x \cdot x+2x \cdot (−4)+3 \cdot x+3 \cdot(−4)&=2x^2-8x+3x-12 \\ &=2x^2−5x-12 \end {aligned}$

Zeros of a Product of Polynomials

Since we made sure that the product of polynomials abides the same laws as if the variables were real numbers, the evaluation of a product of two polynomials in a given point will be the same as the product of the evaluations of the polynomials:

$P(x_0)Q(x_0) = PQ(x_0)$

for all real numbers $x_0.$ 

In particular $PQ(x_0) = 0$ if and only if $P(x_0)Q(x_0)=0$, if and only if $P(x_0) = 0$ or $Q(x_0) = 0$. So the roots of a product of polynomials are exactly the roots of its factors, i.e. $x_0$ is a zero for $PQ(x)$ if it is a zero for $P(x)$ or for $Q(x)$ (and possibly both).