FOIL Shortcut for Multiplying Two Binomials
Example: $(-7f^2+3)(-4f^2-5)$
Warning: FOIL applies to binomials ONLY.
Squaring Binomials
To square a binomial, square the first term, add double the product of the two terms, then add the square of the second term.
That is, $(x+y)^2=x^2+2xy+y^2$
The Most Basic Example: $(a+b)^2$
A More Complex Example: $(4x^2-3p)^2$
$$
\begin{array}{lll}
\displaystyle (\color{blue}{4x^2}\color{red}{-3p})^2&\displaystyle=\left(\color{blue}{4x^2}\right)^2+2(\color{blue}{4x^2})(\color{red}{-3p})+\left(\color{red}{-3p}\right)^2 &\mbox{}\\
\displaystyle &\displaystyle=16x^4-24x^2p+9p^2 &\mbox{exponent rules}\\
\end{array}
$$
DIRE WARNING:
PLEASE REMEMBER...
Differences of Squares
Basic Example $(x+y)(x-y)=$
Example: Multiply by inspection. $(4b-7 z^2)(4b+7 z^2)$
$$
\begin{array}{lll}
\displaystyle (\color{blue}{4b}-\color{red}{7 z^2})(\color{blue}{4b}+\color{red}{7 z^2})&\displaystyle=(\color{blue}{4b})^2-(\color{red}{7 z^2})^2 &\mbox{}\\
\displaystyle &\displaystyle=16b^2-49z^4 &\mbox{exponent rules}\\
\end{array}
$$
Order of Operations
Fact: The order of operations applies to polynomials too.
Example: $(3v - w)(3v + w) - (3v - w)^2$
$$
\begin{array}{lll}
\displaystyle (3v - w)(3v + w) - (3v - w)^2&\displaystyle= (3v)^2-w^2-\left((3v)^2+2(3v)(-w)+(-w)^2\right)&\mbox{}\\
\displaystyle &\displaystyle= 9v^2-w^2-(9v^2-6vw+w^2)&\mbox{}\\
\displaystyle &\displaystyle= 9v^2-w^2-9v^2+6vw-w^2&\mbox{}\\
\displaystyle &\displaystyle=-2w^2+6vw&\mbox{}\\
\end{array}
$$