Expand and simplify: the routine k(a+b) = ka+kb

Expand: distribution

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

$k(a - b) = k a - k b$

The factor in front is distributed to each term inside the parentheses.

Simplify: collect like terms

Like terms share the same variable part.

• $3x$ and $-5x$ are like terms (variable $x$).
• $4x^2$ and $x^2$ are like terms.
• $3x$ and $3x^2$ are not like terms.

Simplify = add coefficients of like terms.

Typical example

Expand and simplify: $3(2x - 4) + 5(x + 1)$.

Distribute: $3 \times 2x - 3 \times 4 + 5 \times x + 5 \times 1 = 6x - 12 + 5x + 5$

Simplify:

• $x$-terms: $6x + 5x = 11x$
• Constant terms: $-12 + 5 = -7$

Result: $11x - 7$.

Watch the signs

If a minus precedes a parenthesis, it flips every sign inside.

$-(3x - 2) = -3x + 2$

Ex: $5x - (2x - 7) = 5x - 2x + 7 = 3x + 7$.

Double distribution

$(a + b)(c + d) = ac + ad + bc + bd$

Each term of the first parenthesis multiplies each term of the second.

Example: $(x + 3)(2x - 1) = 2x^2 - x + 6x - 3 = 2x^2 + 5x - 3$.

Common mistakes

1. Not distributing to every term.
2. Not flipping signs with the − in front of parentheses.
3. Adding unlike terms ($3x + 3x^2 \neq 6x^2$).
4. Writing $(a+b)^2 = a^2 + b^2$ instead of $a^2 + 2ab + b^2$.

?À toi de jouer

Expand and simplify: 2(3x + 4) − (x − 5)?

A.5x + 13B.5x + 3C.7x + 3D.5x − 13

Valider