Introduction to Factorising Brackets

Hello, everyone! Today, we are going to learn about factorising brackets. This is an important skill in maths that helps us simplify expressions and solve equations more easily.

What Does Factorising Mean?

Factorising means breaking down an expression into simpler parts, called factors. When we factorise, we look for numbers or letters that can be multiplied together to give us the original expression.

Why Do We Use Brackets?

Brackets are used in maths to show that certain operations should be done first. When we factorise, we often put factors in brackets to show how they combine to make the original expression.

Key Rules for Factorising Brackets

  1. Look for Common Factors: Always check if there is a number or letter that is common in each term of the expression. For example, in the expression $$2x + 4$$, both terms share a common factor of 2.
  2. Divide by the Common Factor: Once you find the common factor, divide each term by it. For $$2x + 4$$, we divide both terms by 2.
  3. Write in Brackets: After dividing, write the common factor outside the brackets and the remaining terms inside. So, $$2x + 4$$ becomes $$2(x + 2)$$.

Example of Factorising Brackets

Let’s look at an example:

Example 1: Factorise $$6y + 9$$.

  1. Find the Common Factor: The common factor is 3.
  2. Divide: $$6y ÷ 3 = 2y$$ and $$9 ÷ 3 = 3$$.
  3. Write in Brackets: The factorised form is $$3(2y + 3)$$.

Tips for Factorising

  • Always start by looking for the highest common factor.
  • If there are letters involved, check if they share a common variable.
  • Practice makes perfect! The more you factorise, the better you will get.

Practice Questions

Easy Level Questions

  1. Factorise $$4x + 8$$.
  2. Factorise $$3a + 6$$.
  3. Factorise $$5b + 10$$.
  4. Factorise $$2m + 4$$.
  5. Factorise $$7x + 14$$.
  6. Factorise $$9y + 12$$.
  7. Factorise $$10a + 15$$.
  8. Factorise $$6x + 3$$.
  9. Factorise $$8p + 16$$.
  10. Factorise $$4z + 20$$.

Medium Level Questions

  1. Factorise $$12x + 18$$.
  2. Factorise $$15b + 10b$$.
  3. Factorise $$14a + 21$$.
  4. Factorise $$8m + 4n$$.
  5. Factorise $$9x + 27$$.
  6. Factorise $$6y + 15y$$.
  7. Factorise $$10p + 25$$.
  8. Factorise $$5a + 20b$$.
  9. Factorise $$4x + 16y$$.
  10. Factorise $$20k + 15$$.

Hard Level Questions

  1. Factorise $$4x^2 + 8x$$.
  2. Factorise $$6a^2 + 12a$$.
  3. Factorise $$10x^2 + 15x$$.
  4. Factorise $$9y^2 + 27y$$.
  5. Factorise $$5m^2 + 20m + 15$$.
  6. Factorise $$8p^2 + 4p$$.
  7. Factorise $$12x^3 + 6x^2$$.
  8. Factorise $$14a^2 + 21a$$.
  9. Factorise $$16n + 20n^2$$.
  10. Factorise $$24x^2 + 36x$$.

Answers and Explanations

Easy Level Answers

  1. $$4x + 8 = 4(x + 2)$$
    • The common factor is 4. Dividing gives us $$x + 2$$. Thus, we write it as $$4(x + 2)$$.
  2. $$3a + 6 = 3(a + 2)$$
    • The common factor is 3. Dividing gives us $$a + 2$$. Thus, we write it as $$3(a + 2)$$.
  3. $$5b + 10 = 5(b + 2)$$
    • The common factor is 5. Dividing gives us $$b + 2$$. Thus, we write it as $$5(b + 2)$$.
  4. $$2m + 4 = 2(m + 2)$$
    • The common factor is 2. Dividing gives us $$m + 2$$. Thus, we write it as $$2(m + 2)$$.
  5. $$7x + 14 = 7(x + 2)$$
    • The common factor is 7. Dividing gives us $$x + 2$$. Thus, we write it as $$7(x + 2)$$.
  6. $$9y + 12 = 3(3y + 4)$$
    • The common factor is 3. Dividing gives us $$3y + 4$$. Thus, we write it as $$3(3y + 4)$$.
  7. $$10a + 15 = 5(2a + 3)$$
    • The common factor is 5. Dividing gives us $$2a + 3$$. Thus, we write it as $$5(2a + 3)$$.
  8. $$6x + 3 = 3(2x + 1)$$
    • The common factor is 3. Dividing gives us $$2x + 1$$. Thus, we write it as $$3(2x + 1)$$.
  9. $$8p + 16 = 8(p + 2)$$
    • The common factor is 8. Dividing gives us $$p + 2$$. Thus, we write it as $$8(p + 2)$$.
  10. $$4z + 20 = 4(z + 5)$$
  • The common factor is 4. Dividing gives us $$z + 5$$. Thus, we write it as $$4(z + 5)$$.

Medium Level Answers

  1. $$12x + 18 = 6(2x + 3)$$
    • The common factor is 6. Dividing gives us $$2x + 3$$. Thus, we write it as $$6(2x + 3)$$.
  2. $$15b + 10b = 5b(3 + 2)$$
    • The common factor is 5b. Dividing gives us $$3 + 2$$. Thus, we write it as $$5b(3 + 2)$$.
  3. $$14a + 21 = 7(2a + 3)$$
    • The common factor is 7. Dividing gives us $$2a + 3$$. Thus, we write it as $$7(2a + 3)$$.
  4. $$8m + 4n = 4(2m + n)$$
    • The common factor is 4. Dividing gives us $$2m + n$$. Thus, we write it as $$4(2m + n)$$.
  5. $$9x + 27 = 9(x + 3)$$
    • The common factor is 9. Dividing gives us $$x + 3$$. Thus, we write it as $$9(x + 3)$$.
  6. $$6y + 15y = 3y(2 + 5)$$
    • The common factor is 3y. Dividing gives us $$2 + 5$$. Thus, we write it as $$3y(2 + 5)$$.
  7. $$10p + 25 = 5(2p + 5)$$
    • The common factor is 5. Dividing gives us $$2p + 5$$. Thus, we write it as $$5(2p + 5)$$.
  8. $$5a + 20b = 5(1a + 4b)$$
    • The common factor is 5. Dividing gives us $$1a + 4b$$. Thus, we write it as $$5(1a + 4b)$$.
  9. $$4x + 16y = 4(x + 4y)$$
    • The common factor is 4. Dividing gives us $$x + 4y$$. Thus, we write it as $$4(x + 4y)$$.
  10. $$20k + 15 = 5(4k + 3)$$
    • The common factor is 5. Dividing gives us $$4k + 3$$. Thus, we write it as $$5(4k + 3)$$.

Hard Level Answers

  1. $$4x^2 + 8x = 4x(x + 2)$$
    • The common factor is 4x. Dividing gives us $$x + 2$$. Thus, we write it as $$4x(x + 2)$$.
  2. $$6a^2 + 12a = 6a(a + 2)$$
    • The common factor is 6a. Dividing gives us $$a + 2$$. Thus, we write it as $$6a(a + 2)$$.
  3. $$10x^2 + 15x = 5x(2x + 3)$$
    • The common factor is 5x. Dividing gives us $$2x + 3$$. Thus, we write it as $$5x(2x + 3)$$.
  4. $$9y^2 + 27y = 9y(y + 3)$$
    • The common factor is 9y. Dividing gives us $$y + 3$$. Thus, we write it as $$9y(y + 3)$$.
  5. $$5m^2 + 20m + 15 = 5(m^2 + 4m + 3)$$
    • The common factor is 5. Dividing gives us $$m^2 + 4m + 3$$. Thus, we write it as $$5(m^2 + 4m + 3)$$.
  6. $$8p^2 + 4p = 4p(2p + 1)$$
    • The common factor is 4p. Dividing gives us $$2p + 1$$. Thus, we write it as $$4p(2p + 1)$$.
  7. $$12x^3 + 6x^2 = 6x^2(2x + 1)$$
    • The common factor is 6x². Dividing gives us $$2x + 1$$. Thus, we write it as $$6x^2(2x + 1)$$.
  8. $$14a^2 + 21a = 7a(2a + 3)$$
    • The common factor is 7a. Dividing gives us $$2a + 3$$. Thus, we write it as $$7a(2a + 3)$$.
  9. $$16n + 20n^2 = 4n(4 + 5n)$$
    • The common factor is 4n. Dividing gives us $$4 + 5n$$. Thus, we write it as $$4n(4 + 5n)$$.
  10. $$24x^2 + 36x = 12x(2x + 3)$$
    • The common factor is 12x. Dividing gives us $$2x + 3$$. Thus, we write it as $$12x(2x + 3)$$.

I hope this helps you understand factorising brackets better! Keep practicing, and soon it will become second nature. If you have any questions, feel free to ask!