When multiplying two algebraic expressions, it may be helpful if you remember FOIL.
F First Terms
O Outer Terms
I Inner Terms
L Last Terms


In the figure above,
\(A \times C\) are First Terms.
\(A \times D\) are Outer Terms.
\(B \times C\) are Inner Terms.
\(B \times D\) are Last Terms.

Example Expand \((3y+1)(3x+2y)\).
 Solution  \((3y+1)(3x+2y) \\ \, \\ =\underbrace{(3y\cdot3x)}_{\text{First Terms}}+\underbrace{(3y\cdot2y)}_{\text{Outer Terms}}+\underbrace{(1\cdot3x)}_{\text{Inner Terms}}+\underbrace{(1\cdot2y)}_{\text{Last Terms}} \\ \, \\ =9xy+6y^2+3x+2y\)

Expand \((4x+1)(2x+3y)\).
Answer
\((4x+1)(2x+3y) \\ \, \\ =\underbrace{(4x\cdot2x)}_{\text{First Terms}}+\underbrace{(4x\cdot3y)}_{\text{Outer Terms}}+\underbrace{(1\cdot2x)}_{\text{Inner Terms}}+\underbrace{(1\cdot3y)}_{\text{Last Terms}} \\ \, \\ =8x^2+12xy+2x+3y \)

Expand \((2+5y)(x+7y)\).
Answer
\((2+5y)(x+7y) \\ \, \\ =\underbrace{(2\cdot x)}_{\text{First Terms}}+\underbrace{(2\cdot7y)}_{\text{Outer Terms}}+\underbrace{(5y\cdot x)}_{\text{Inner Terms}}+\underbrace{(5y\cdot7y)}_{\text{Last Terms}} \\ \, \\ =2x+14y+5xy+35y^2 \)

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Special Rules

There are three types of patterns when multiplying terms. It may be a good idea to memorize these, as they are quite common algebraic expressions that appear in the test.

\(1. \quad (A + B)^2=A^2+2AB+B^2\)

Example Expand \((3x+2y)^2\).
 Solution  \((3x+2y)^2 \\ \, \\ =\underbrace{(3x)^2}_{A^2}+\underbrace{2(3x)(2y)}_{2AB}+\underbrace{(2y)^2}_{B^2} \\ \, \\ =9x^2+12xy+4y^2 \)


\(2. \quad (A – B)^2=A^2-2AB+B^2\)

Example Expand \((3x−2y)^2\).
 Solution  \((3x−2y)^2 \\ \, \\ =\underbrace{(3x)^2}_{A^2}-\underbrace{2(3x)(2y)}_{2AB}+\underbrace{(2y)^2}_{B^2} \\ \, \\ =9x^2-12xy+4y^2 \)


\(3. \quad (A+B)(A-B)=A^2-B^2\)

Example Expand \((3x+2y)(3x−2y)\).
 Solution  \((3x+2y)(3x−2y) \\ \, \\ =\underbrace{9x^2}_{A^2}-\underbrace{4y^2}_{B^2} \)


Expand \((4x+y)(4x−y)\).
Answer
\((4x+y)(4x−y) \\ \, \\ =\underbrace{(4x)^2}_{A^2}-\underbrace{(y)^2}_{B^2} \\ \, \\ =16x^2-y^2\)

Note: This is Type 3, \((A+B)(A-B)=A^2-B^2\)


Expand \((8x+y)^2\).
Answer
\((8x+y)^2 \\ \, \\ =\underbrace{(8x)^2}_{A^2}+\underbrace{2(8x)(y)}_{2AB}+\underbrace{(y)^2}_{B^2} \\ \, \\ =64x^2+16xy+y^2\)

Note: This is Type 1, \((A + B)^2=A^2+2AB+B^2\)


Expand \((2k-5)^2\).
Answer
\((2k-5)^2 \\ \, \\ =\underbrace{(2k)^2}_{A^2}-\underbrace{2(2k)(5)}_{2AB}+\underbrace{(5)^2}_{B^2} \\ \, \\ =4k^2-20k+25\)

Note: This is Type 2, \((A – B)^2=A^2-2AB+B^2\)

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