SIGN FLIP
Negative outside → ALL signs inside flip!
−(a+b) = −a−b
FOIL
First · Outer · Inner · Last
(a+b)(c+d) order
DISTRIBUTE
Multiply EVERY term inside
a(b+c) = ab+ac
LIKE TERMS
Collect same-power terms last
3x+5x = 8x ✓
NEG × NEG
Negative × Negative = POSITIVE
−3 × −x = +3x
EXPAND FIRST
Always expand before simplifying
Never skip steps!
1 Basic Distribution (Warm-Up)
EXAMPLE
WORKED EXAMPLE · Monomial × Binomial
Expand: \( 3x(2x - 5) \)
1Distribute \(3x\) to each term: \(3x \cdot 2x\) and \(3x \cdot (-5)\)
2\( = 6x^2 - 15x \)
→ multiply EVERY term! don't miss the second one
Watch the sign! \(-3x(x - 4) = -3x^2 \mathbf{+} 12x\) — the minus × minus = plus!
2 FOIL — Binomial × Binomial
EXAMPLE
WORKED EXAMPLE · FOIL Method
Expand: \( (x + 3)(x - 2) \)
\( \underbrace{x \cdot x}_{\text{First}} + \underbrace{x \cdot (-2)}_{\text{Outer}} + \underbrace{3 \cdot x}_{\text{Inner}} + \underbrace{3 \cdot (-2)}_{\text{Last}} \)
1\( = x^2 - 2x + 3x - 6 \)
2Collect like terms: \( = x^2 + x - 6 \)
The middle terms \(-2x + 3x\) combine — don't forget this step!
3 Special Patterns
EXAMPLE
WORKED EXAMPLE · Perfect Square & Difference of Squares
Perfect Square: \( (a+b)^2 = a^2 + 2ab + b^2 \)
e.g. \( (x+3)^2 = x^2 + 6x + 9 \)
Diff. of Squares: \( (a+b)(a-b) = a^2 - b^2 \)
e.g. \( (x+5)(x-5) = x^2 - 25 \)
\((x+3)^2 \neq x^2 + 9\) — you MUST include the middle term \(+6x\)!
4 Negative Outside Brackets
EXAMPLE
WORKED EXAMPLE · Negative Distribution
Expand: \( -3(x^2 - 2x + 4) \)
1Distribute \(-3\) to ALL terms
2\( -3 \cdot x^2 = -3x^2 \)
3\( -3 \cdot (-2x) = \mathbf{+6x} \) ← sign flips!
4\( -3 \cdot 4 = -12 \)
Answer: \( -3x^2 + 6x - 12 \)
5 Mixed Challenges