Expanding Single Brackets
BGB-360 ACADEMY | SECTION 2
Expanding Single Brackets
Multiply every term inside the brackets, then simplify your result.
Your Learning Targets
- Explain and apply the distributive law.
- Expand brackets with positive and negative multipliers.
- Expand brackets with a variable outside.
- Collect like terms after expanding.
1. What Does Expanding Mean?
Expanding means multiplying out brackets to write an equivalent expression without those brackets.
For example, 3(x + 4) means three copies of the whole expression x + 4:
(x + 4) + (x + 4) + (x + 4) = 3x + 12
We can obtain the same result by multiplying each term inside by three.
The Distributive Law
a(b + c) = ab + ac
a(b − c) = ab − ac
The multiplier outside applies to every term inside the brackets.
2. A Positive Multiplier
Example A: Addition Inside
Expand 4(x + 3).
= 4 × x + 4 × 3
= 4x + 12
Example B: Subtraction Inside
Expand 3(2y − 5).
= 3 × 2y + 3 × (−5)
= 6y − 15
Your turn: Expand 5(2a + 3). Attempt it before reading the check.
Check: 5 × 2a + 5 × 3 = 10a + 15.
3. A Negative Multiplier
Keep the sign of the outside multiplier with it. Use the multiplication sign rules:
- Negative × positive = negative.
- Negative × negative = positive.
Example C: Both Signs Matter
Expand −2(3x − 4).
= (−2)(3x) + (−2)(−4)
= −6x + 8
The second term becomes positive because two negative factors multiply to give a positive result.
Your turn: Expand −3(2p + 5).
Check: (−3)(2p) + (−3)(5) = −6p − 15.
4. A Minus Sign Before Brackets
A minus sign immediately before brackets means multiply the whole bracket by −1.
−(x + 4) = −x − 4
−(x − 4) = −x + 4
Example D: Subtract a Whole Expression
Simplify 7x − (2x − 3).
Multiply each bracket term by −1:
= 7x − 2x + 3
= 5x + 3
Your turn: Simplify 8a − (3a + 2).
Check: 8a − 3a − 2 = 5a − 2.
5. A Variable Outside the Brackets
The same distributive law applies when the multiplier contains a variable.
Remember: x × x = x2.
Example E: Multiply Coefficients and Variables
Expand 2x(3x − 4).
= (2x)(3x) + (2x)(−4)
= 6x2 − 8x
For the first term, 2 × 3 = 6 and x × x = x2.
Your turn: Expand 3y(y + 2).
Check: 3y2 + 6y.
6. Expand, Then Collect Like Terms
When an expression contains several brackets, expand each one before combining matching terms.
Example F: Two Separate Brackets
Simplify 3(x + 2) + 2(x − 5).
Expand:
= 3x + 6 + 2x − 10
Collect like terms:
= 5x − 4
Example G: Subtracting a Bracket Product
Simplify 4(2a − 1) − 3(a + 2).
Treat the second multiplier as −3:
= 8a − 4 − 3a − 6
= 5a − 10
Your turn: Simplify 2(3t + 4) − (t − 5).
Check: 6t + 8 − t + 5 = 5t + 13.
7. Check Your Expansion
Check −2(3x − 4) = −6x + 8 using x = 2:
- Original: −2(6 − 4) = −2(2) = −4.
- Expanded: −6(2) + 8 = −12 + 8 = −4.
A numerical check can reveal mistakes, but matching at one value does not prove equivalence for every value. Your expansion must follow the distributive law.
Avoid These Common Mistakes
- Multiplying only the first term: 3(x + 4) = 3x + 12, not 3x + 4.
- Losing a negative sign: −2(x + 3) = −2x − 6.
- Subtracting only one bracket term: −(x − 5) = −x + 5.
- Forgetting variable multiplication: x(x + 2) = x2 + 2x.
- Combining unlike terms: 6x2 − 8x cannot be collected into one term.
Pause and Reflect
Complete these statements in your notebook:
The outside multiplier must multiply...
A minus sign before brackets means...
After expanding, I should check for...
Next: Expanding Double Brackets
Use the distributive law again to multiply every term in one bracket by every term in the other.
