# Multiplication of Polynomials by Binomials

## Key Questions

It's a rule.

#### Explanation:

It's a rule commonly used in factoring, meaning to start by multiplying the two first variables first, then outer, then inner, then last.
Ex:
If the things being multiplied is (x+1) by (x-2), you would multiply "x" and "x" first.
$x \cdot x = {x}^{2}$
$x \cdot - 2 = - 2 x$
$1 \cdot x = x$
$1 \cdot - 2 = - 2$
The final answer would be: ${x}^{2} - x - 2$

• The distribution property says that $a \cdot \left(b + c\right) = a \cdot b + a \cdot c$

With more polynomials it gets a bit harder. I'll do it the long way:

$\left(a + b\right) \cdot \left(c + d\right) = \left(a + b\right) \cdot c + \left(a + b\right) \cdot d$

We have distributed the second binomial, and we now distribute the first binomial (twice):

$\left(a + b\right) \cdot c + \left(a + b\right) \cdot d = a \cdot c + b \cdot c + a \cdot d + b \cdot d$

With larger polynomials the 'book-keeping' may become a bit tedious, and most trained people take shortcuts.

If you have more than two polynomials, best method is to do them step by step, two at a time:

$\left(a + b\right) \left(c + d\right) \cdot \left(e + f\right)$

$= \left(a c + a d + b c + b d\right) \left(e + f\right)$ (see above)

$= a c e + a c f + a \mathrm{de} + a \mathrm{df} + b c e + b c f + b \mathrm{de} + b \mathrm{df}$

Last check: 2-term times 2-term = 4 terms
4-terms times 2-term = 8-terms.
In practical examples, you will be able to add like terms (like the numbers, $x$'s ${x}^{2}$'s, etc.
(there are no like terms in this example)