# Multiplication of Monomials by Polynomials

## Key Questions

• It works the same as with numbers. For numbers, you know that $a \left(b + c\right)$ equals $a b + a c$.
For the same reason, if you have a monomial and you want to multiplicate it by a polynomial (which is a sum of monomials with some coefficients!), you follow the same rule.

For example, if your monomial is $3 {x}^{2}$, and your polynomial is $3 + 2 x - 5 {x}^{2} + 8 {x}^{3}$, the product is
$3 {x}^{2} \left(3 + 2 x - 5 {x}^{2} + 8 {x}^{3}\right)$
you will calculate is as
$3 {x}^{2} \setminus \cdot 3 + 3 {x}^{2} \setminus \cdot 2 x - 3 {x}^{2} \setminus \cdot 5 {x}^{2} + 3 {x}^{2} \setminus \cdot 8 {x}^{3}$, which is
$9 {x}^{2} + 6 {x}^{3} - 15 {x}^{4} + 24 {x}^{5}$

$\implies {a}_{1} {x}^{{p}_{1}} \cdot {a}_{2} {x}^{{p}_{2}} = {a}_{1} {a}_{2} {x}^{{p}_{1} + {p}_{2}}$

#### Explanation:

A monomial is of the form:

$\implies a {x}^{p}$

where $a$ is a constant coefficient and $p$ is a constant power.

In the case of multiplying two monomials together:

$\implies A {x}^{P} \equiv {a}_{1} {x}^{{p}_{1}} \cdot {a}_{2} {x}^{{p}_{2}}$

The coefficients will multiply, so:

$\implies A = {a}_{1} \cdot {a}_{2}$

The powers will sum, so:

$\implies P = {p}_{1} + {p}_{2}$

Hence:

$\implies A {x}^{P} \equiv {a}_{1} {x}^{{p}_{1}} \cdot {a}_{2} {x}^{{p}_{2}} = {a}_{1} {a}_{2} {x}^{{p}_{1} + {p}_{2}}$

For example:
$\implies 3 {x}^{2} \cdot 2 x$

$\implies \left(3 \cdot 2\right) {x}^{2 + 1}$

$\implies 6 {x}^{3}$

• Just distribute the monomial to each of the polynomial's terms

For example:

$\left(3 m\right) \left({m}^{2} - 2 m + 1\right)$

$\implies \left(3 m\right) \left({m}^{2}\right) - \left(3 m\right) \left(2 m\right) + \left(3 m\right) \left(1\right)$
$\implies 3 {m}^{3} - 6 {m}^{2} + 3 m$