Distributive property: expand without ever getting it wrong

The rule

$k(a + b) = ka + kb$

The factor in front distributes to each term inside the parentheses.

The key move

Mentally (or literally) draw two arrows from the factor to each term. Miss one = wrong.

$3(x + 5) = \underbrace{3 \times x}_{\text{arrow 1}} + \underbrace{3 \times 5}_{\text{arrow 2}} = 3x + 15$

Watch the minus sign

$-4(x - 2) = -4x + 8$ (the − becoming + is trap #1).

Safe method: treat the factor as a “signed package” ($-4$) and distribute.

Double distributivity: $(a+b)(c+d)$

Each term of the first multiplies each term of the second. 4 products.

$(x + 3)(x + 2) = x \cdot x + x \cdot 2 + 3 \cdot x + 3 \cdot 2 = x^2 + 5x + 6$.

Classic traps

• $3(x+2) \ne 3x + 2$ (forgetting the second product).
• $-(x-5) = -x + 5$, not $-x - 5$.

?Your turn

Expanding −5(2x − 3) gives:

A.−10x − 15B.−10x + 15C.10x − 15D.10x + 15

Check