## Factoring Using FOIL

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

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 $$x$$.
$$x^2 -x = 20$$
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 $$x$$.
$$x^2 +x -4 = -3x + 17$$
$$x^2 +x -4 {\color{red}{+3x-17}} = 20 {\color{red}{+3x-17}} \\ \, \\ x^2 +4x -21 = 0$$
$$x^2 +4x -21 = \underbrace{(x+ \bigcirc )}_{\substack{\text{Think} \\ \text{possible} \\ \text{value} \\ \text{in blank}}} \cdot \underbrace{(x+ \bigcirc )}_{\substack{\text{Think} \\ \text{possible} \\ \text{value} \\ \text{in blank}}} \\ \, \\ =(x-3)(x+7)$$
$$(x-3)=0$$ or $$(x+7)=0$$.
$$x=3$$ or $$x=-7$$