In Grade 6 you used the distributive property: $3(x + 2) = 3x + 6$. Now the numbers can be negative or fractions, and you will also run it backward to factor.
Key idea
Expand: multiply the number outside by every term inside: $a(b + c) = ab + ac$. Factor: go the other way and pull out a common factor: $12x - 18 = 6(2x - 3)$. Check a factoring by expanding it again.
Why it works
A rectangle that is 3 wide and $x + 4$ long can be cut into two pieces, $3 \times x$ and $3 \times 4$. Its area is $3(x + 4)$ and also $3x + 12$. A minus sign in front of parentheses means $-1$ times the group, so $-(x - 4) = (-1)(x) + (-1)(-4) = -x + 4$: every sign inside flips.
Worked examples
Worked example: Expanding with a negative
Expand $-3(2x - 5)$.
- $-3 \times 2x = -6x$.
- $-3 \times (-5) = 15$.
- Together: $-6x + 15$.
Answer: $-6x + 15$
Worked example: Factoring
Factor $12x - 18$ using the greatest common factor.
- The greatest common factor of 12 and 18 is 6.
- $12x \div 6 = 2x$ and $-18 \div 6 = -3$, so $12x - 18 = 6(2x - 3)$.
- Check: $6(2x - 3) = 12x - 18$ ✓.
Answer: $6(2x - 3)$
Worked example: Factoring out a negative
Fill in the box: $-8x + 6 = -2(\square)$.
- Divide each term by $-2$: $-8x \div (-2) = 4x$ and $6 \div (-2) = -3$.
- So $-8x + 6 = -2(4x - 3)$. Check: $-2 \times 4x = -8x$ and $-2 \times (-3) = 6$ ✓.
Answer: $4x - 3$
Common mistake
Distributing to the first term only: $-(x - 4) = -x - 4$. The $-1$ multiplies both terms, so the answer is $-x + 4$. Check with $x = 10$: $-(10 - 4) = -6$, and $-10 + 4 = -6$ ✓, but $-10 - 4 = -14$ ✗.