Factoring Using FOIL

Do you remember FOIL?
F First Terms
O Outer Terms
I Inner Terms
L Last Terms

  Review here  if you don’t remember it.

Example Find factors from \(x^2+4x+3\).
 Solution 
\(x^2+4x+3= \underbrace{(x+ \bigcirc )}_{\substack{\text{Think} \\ \text{possible} \\ \text{value} \\ \text{in blank}}} \cdot \underbrace{(x+ \bigcirc )}_{\substack{\text{Think} \\ \text{possible} \\ \text{value} \\ \text{in blank}}} \\ \, \\ = (x+3)(x+1)\)


Solving by Factoring

To solve equations by factoring,
  1. Bring ALL expressions to one side.
  2. Then, the other side will be 0.
  3. Factor into a product of expressions.

Example Solve for \(x\).
\(x^3-2x^2+x=-5(x-1)^2\)

 Solution  Bring all expressions to one side.
\(x^3-2x^2+x {\color{red}{+5(x-1)^2}}=-5(x-1)^2{\color{red}{+5(x-1)^2}} \\ \, \\ x^3-2x^2+x+5(x-1)^2=0 \)

Factor.
\(x(x^2-2x+1)+5(x-1)^2=0 \\ \, \\ x\underbrace{(x-1)(x-1)}_{\substack{\text{Factor using} \\ \text{FOIL} }}+5(x-1)^2=0 \\ \, \\ x(x-1)^2 +5(x-1)^2=0 \\ \, \\ x\underbrace{{\color{red}{(x-1)^2}}}_{\substack{\text{Common} \\ \text{Factor} }} +5\underbrace{{\color{red}{(x-1)^2}}}_{\substack{\text{Common} \\ \text{Factor} }}=0 \\ \, \\ (x+5)(x-1)^2=0 \\ \, \\ \)
\((x+5)=0\) or \((x-1)^2=0\).
\(x=-5\) or \(x=1\)

Solve for \(x\).
\(x^2 -x = 20\)
Answer
Bring all expressions to one side.
\(x^2 -x {\color{red}{-20}} = 20 {\color{red}{-20}} \\ \, \\ x^2 -x – 20 = 0 \)

Factor using FOIL.
\( x^2 -x – 20 = \underbrace{(x+ \bigcirc )}_{\substack{\text{Think} \\ \text{possible} \\ \text{value} \\ \text{in blank}}} \cdot \underbrace{(x+ \bigcirc )}_{\substack{\text{Think} \\ \text{possible} \\ \text{value} \\ \text{in blank}}} \\ \, \\ =(x+4)(x-5) \)

\((x+4)=0\) or \((x-5)=0\).
\(x=-4\) or \(x=5\)

Solve for \(x\).
\(x^2 +x -4 = -3x + 17\)
Answer
Bring all expressions to one side.
\(x^2 +x -4 {\color{red}{+3x-17}} = 20 {\color{red}{+3x-17}} \\ \, \\ x^2 +4x -21 = 0 \)

Factor using FOIL.
\( x^2 +4x -21 = \underbrace{(x+ \bigcirc )}_{\substack{\text{Think} \\ \text{possible} \\ \text{value} \\ \text{in blank}}} \cdot \underbrace{(x+ \bigcirc )}_{\substack{\text{Think} \\ \text{possible} \\ \text{value} \\ \text{in blank}}} \\ \, \\ =(x-3)(x+7) \)

\((x-3)=0\) or \((x+7)=0\).
\(x=3\) or \(x=-7\)
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