What is a binomial?
A binomial is an algebraic expression of the sum (+) or the difference (-) of two terms.
Quick Review:
Let’s review some terms and expressions that might help us understand these.
Monomial examples: $5y, 8x^2, -2x$
Binomial examples: $-3x^2-2,9y-2y^2$
Polynomial examples: $8x^2+3x-2,12x^2+11x+2$
What is a polynomial?
Polynomials are algebraic expressions that include real numbers (positive, negative, large, small, whole, or decimal numbers) and variables (x, y, etc.). They include more than term and are the sum of monomials. They are usually also written in decreasing order of terms.
| Term | Polynimal or Not? | Why? |
|---|---|---|
| $8x^2+3x-2$ | Polynomial | |
| $8x^{−3}-7y-2$ | NOT a polynomial | The exponent is negative ($x^{−3}$) |
| $8x^2+8x-\frac{2}{3}$ | NOT a polynomial | Cannot have division |
Now...how do we multiply binomials?
When multiplying binomials, you can use the FOIL method. For instance, to find the product of 2 binomials, you’ll add the products of the First terms, the Outer terms, the Inner terms, and the Last terms.
FIRST: multiply the first term in each set of parenthesis
OUTER: multiply the outer term in each set of parenthesis
INNER: multiply the inner term in each set of parenthesis
LAST: multiply the last term in each set of parenthesis
Example 1
Let’s work out this problem.
$$(3x-7)(5x+6)$$
1. Identify the FOIL numbers
\begin{align*}
\color{#FF0000}{\text{First}}&:3x\cdot5x\\
\color{#FF8C00}{\text{Outer}}&:3x\cdot6\\
\color{#008000}{\text{Inner}}&:(−7)\cdot5x\\
\color{black}{\text{Last}}&:(−7)\cdot6\\
\end{align*}
2. Multiply the terms
\begin{align*}
\color{#FF0000}{\text{First}}&:3x\cdot5x=\color{#FF0000}{15x^2}\\
\color{#FF8C00}{\text{Outer}}&:3x\cdot6=\color{#FF8C00}{18x}\\
\color{#008000}{\text{Inner}}&:(−7)\cdot5x=\color{#008000}{−35x}\\
\color{black}{\text{Last}}&:(−7)\cdot6=\color{black}{−42}\\
\end{align*}
3. Combine like terms
\begin{align*}
$\color{#FF8C00} {18x} \color{#008000} {−35x}= {−17x}
\end{align*}
4. Combine all terms in decreasing order
\begin{align*}
$\color{#FF0000} {15x^2} \color{#FF8C00} {−17x} \color{black} {−42}
\end{align*}
Final Answer: $$15x^2-17x-42$$
Example 2
$$(5+4x)(3+2x)$$
$$(5+4x)(3+2x)$$
$$= \color{#FF0000}{(5\cdot3)} + \color{#FF8C00}{(5\cdot2x)} + \color{#008000}{(4x\cdot3)} + \color{black}{(4x\cdot2x)}$$
$$= \color{#FF0000}{15} + \color{#FF8C00}{10x} + \color{#008000}{12x} + \color{black}{8x^2}$$
$$= \color{#FF0000}{15} + \color{#FF8C00}{22x}+ \color{black}{8x^2}$$
$$= \color{black}{8x^2}+ \color{#FF8C00}{22x}+ \color{#FF0000}{15}$$
Final answer: $$8x^2+22x+15$$