What does it mean?
The distributive property multiplies every part of a grouped sum. Three copies of (2x + 5) contain six x-units and fifteen ordinary units. Collecting like terms then combines quantities of the same algebraic type: 3x and 2x can become 5x, but 3x and 2 cannot.
A bracket groups terms that are multiplied by the same number. 5(x + 2) means five copies of x and five copies of 2, so it expands to 5x + 10. Multiply the outside term by every term inside, including the sign of each inside term. After expanding, collect like terms: x terms with x terms, and numbers with numbers. 3x and 4 are unlike, so they stay separate.
Learning goal: Expand a single bracket and collect like terms. The grade is a suggested starting point; school sequences vary. Use the first worked example to check readiness, and Pre-algebra if an earlier idea needs revisiting.
See the idea
Read each diagram with its labels, then follow the worked examples. Practice questions include their own measurements and information.
Enlarge diagram to read the labels
Swipe across the enlarged diagram to read every label.
Enlarge diagram to read the labels
Swipe across the enlarged diagram to read every label.
Choose a method and explain why it works
Draw an arrow from the outside multiplier to every term, including its sign. Write the full expansion on a separate line before collecting terms. When a minus precedes a bracket, treat it as multiplying by −1. Check an expansion by substituting a small value into the original and expanded forms.
Notation: 3(2x − 5) = 6x − 15. A minus outside, −2(x + 4), changes both signs: −2x − 8.
Key rule and when to use it
a(b + c) = ab + ac. Like terms have exactly the same variable part, including powers. A minus before brackets acts as multiplication by −1.
Worked examples, step by step
Example 1: Expand 3(2x + 5).
- Multiply 3 by 2x and 3 by 5.
- 3 × 2x = 6x and 3 × 5 = 15.
- Write 6x + 15.
Answer: 6x + 15
Check: If x = 1, the bracket is 7 and 3 × 7 = 21. 6(1) + 15 = 21, so the expansion matches.
Example 2: Expand and simplify 4x − 2(x − 3).
- This is not a single product. First expand the bracket with the minus 2: −2(x − 3) = −2x + 6.
- The expression is now 4x − 2x + 6.
- Collect like x terms: 2x + 6.
Answer: 2x + 6
Check: If x = 4, original: 16 − 2(1) = 14. Simplified: 8 + 6 = 14.
Example 3: Expand and simplify 5(x − 2) − 2(x + 3).
- Expand both groups: 5x − 10 − 2x − 6.
- Collect x-terms and constants: 3x − 16.
- At x = 2, the original gives 0 − 10 = −10; the simplified form gives 6 − 16 = −10.
Common mistakes and important boundaries
x and x² are unlike terms, just as lengths and areas are different measurements. Do not combine them by adding coefficients. In −2(x − 3), the second product is positive six because −2 × −3 = 6. Losing that sign changes the expression for almost every x.
A point to explain
A learner may multiply only the first term inside a bracket, or add unlike terms such as 3x and 4.
Compare an uncertain attempt with the closest worked example. Look at the meaning of each operation, unit and label before repeating the arithmetic.
Expanding Brackets and Collecting Like Terms — practice
Work on paper first. Open each answer afterwards and compare the reasoning. Print the questions with or without the separate answer key; no account is required.
Name: ____________________ Date: __________
What does 5(x + 2) mean?
Answer and a way to check
Five groups of x and five groups of 2
The 5 multiplies every term inside the bracket, not only one of them.
Expand 2(3x + 4).
Answer and a way to check
6x + 8
Multiply 2 by 3x and 2 by 4. Write both products.
Expand 6(x − 5).
Answer and a way to check
6x − 30
6 multiplies x and 6 multiplies −5. Keep the minus with the second product.
Collect like terms: 3x + 2x − 7 + 4.
Answer and a way to check
5x − 3
Join the x terms. Then join the numbers. Do not add 5x to −3.
Expand and simplify 4(2x + 1) + 3x.
Answer and a way to check
11x + 4
Expand the bracket first. Then add 3x to the x term only.
Explain why 2(x + 5) is not equal to 2x + 5.
Answer and a way to check
2 multiplies both x and 5, so the expansion is 2x + 10, not 2x + 5.
Check with a number. Try x = 1 in both writings and compare.
A student writes 3(x − 4) = 3x − 4. What is the mistake?
Answer and a way to check
The student did not multiply 3 by −4.
The 3 must multiply both terms. What is 3 × (−4)?
A rectangle has width 3 cm and length (x + 7) cm. Write a simplified expression for its area.
Answer and a way to check
3x + 21
Area is width × length. Expand 3(x + 7).
Expanding Brackets and Collecting Like Terms — answer key
Five groups of x and five groups of 2 The 5 multiplies every term inside the bracket, not only one of them.
6x + 8 Multiply 2 by 3x and 2 by 4. Write both products.
6x − 30 6 multiplies x and 6 multiplies −5. Keep the minus with the second product.
5x − 3 Join the x terms. Then join the numbers. Do not add 5x to −3.
11x + 4 Expand the bracket first. Then add 3x to the x term only.
2 multiplies both x and 5, so the expansion is 2x + 10, not 2x + 5. Check with a number. Try x = 1 in both writings and compare.
The student did not multiply 3 by −4. The 3 must multiply both terms. What is 3 × (−4)?
3x + 21 Area is width × length. Expand 3(x + 7).
Questions students and parents ask
Can I collect 2xy and 3yx?
Yes. Multiplication commutes, so xy = yx and the sum is 5xy. But 2x and 3x² have different variable parts and cannot be collected into one like term.
Is 3x + 4 already simplified?
Yes, if there are no further operations to perform. The terms are unlike, so combining them into 7x would change the value.
Is this a complete set of textbook exercise solutions?
No. This guide adapts BananaLearn’s own teaching examples and practice. It is not official NCERT material or a promise of the exact chapter sequence in a particular board, country or textbook edition.
What if the examples are still difficult?
Revisit Pre-algebra. Try the first example with its model, then attempt one fresh question independently. A correct copied answer is not evidence that the method can be used without support.
Help at home
Have the learner mark term signs before expanding. Ask them to explain why each outside factor must act on the complete bracket.
Keep one example of the learner’s own working. A repeated difficulty across fresh questions can guide a discussion with their teacher; one short practice sheet does not establish a diagnosis or a full mastery profile.
Connect this idea to the next skill
About this guide
Based on BananaLearn’s Grade 8 course lesson “Expand and Collect”. The explanations have been adapted for independent reading; course diagrams are reused where they fit the idea, and missing classroom givens are made explicit in questions.
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