Introduction to Factors

Hello, Year 7! Today, we’re going to learn about factors of linear expressions. A linear expression is an expression that can be written in the form of $$ax + b$$, where $$a$$ and $$b$$ are numbers, and $$x$$ is a variable.

What are Factors?

Factors are numbers or expressions that can be multiplied together to get another number or expression. For example, in the expression $$6x$$, the factors could be $$2$$, $$3$$, and $$x$$ because $$2 \times 3 \times x = 6x$$.

Why is Factoring Important?

Factoring helps us simplify expressions and solve equations. It’s like breaking down a big number or expression into smaller, more manageable parts. This is useful in solving problems and understanding how different parts of an expression work together.

Key Rules for Factoring Linear Expressions

  1. Look for Common Factors: Always check if there’s a number or variable that is common in all terms of the expression.Example: In the expression $$4x + 8$$, the common factor is $$4$$. Factoring it gives us $$4(x + 2)$$.
  2. Use the Distributive Property: This property states that $$a(b + c) = ab + ac$$. You can use it to factor expressions back into a product.Example: If we have $$3x + 6$$, we can factor out $$3$$ to get $$3(x + 2)$$.
  3. Keep it Simple: Start with the simplest numbers first. Sometimes, just pulling out a common number can make things clearer.

Tips and Tricks

  • Practice Makes Perfect: The more you practice factoring, the better you will get at it! Try breaking down expressions into factors in different ways.
  • Draw it Out: Sometimes, a visual can help! Drawing a number line or using blocks can help you see how factors work.
  • Check Your Work: After factoring, you can always multiply your factors back together to see if you get the original expression.

Examples

  1. Example 1: Factor the expression $$2x + 4$$.
    • Common factor is $$2$$.
    • Factored form is $$2(x + 2)$$.
  2. Example 2: Factor the expression $$5x + 10$$.
    • Common factor is $$5$$.
    • Factored form is $$5(x + 2)$$.
  3. Example 3: Factor the expression $$7x – 14$$.
    • Common factor is $$7$$.
    • Factored form is $$7(x – 2)$$.

Practice Questions

Easy Level Questions

  1. Factor $$3x + 6$$.
  2. Factor $$8y + 16$$.
  3. Factor $$5a + 10$$.
  4. Factor $$4x – 8$$.
  5. Factor $$12y + 24$$.
  6. Factor $$10x + 20$$.
  7. Factor $$6x – 12$$.
  8. Factor $$9a + 18$$.
  9. Factor $$2x + 10$$.
  10. Factor $$7b + 14$$.
  11. Factor $$15m + 30$$.
  12. Factor $$20y – 40$$.
  13. Factor $$18x + 36$$.
  14. Factor $$11n + 22$$.
  15. Factor $$13p – 39$$.
  16. Factor $$6k + 12$$.
  17. Factor $$2x – 4$$.
  18. Factor $$4y + 8$$.
  19. Factor $$5x + 15$$.
  20. Factor $$2a + 4$$.

Medium Level Questions

  1. Factor $$x^2 + 2x$$.
  2. Factor $$3x^2 + 6x$$.
  3. Factor $$2x^2 – 4x$$.
  4. Factor $$4x^2 + 8x$$.
  5. Factor $$5x^2 – 10x$$.
  6. Factor $$6y^2 + 12y$$.
  7. Factor $$8m^2 + 16m$$.
  8. Factor $$9n^2 – 27n$$.
  9. Factor $$7k^2 + 14k$$.
  10. Factor $$2x^2 + 10x$$.
  11. Factor $$4a^2 – 8a$$.
  12. Factor $$3b^2 + 12b$$.
  13. Factor $$5p^2 – 15p$$.
  14. Factor $$6x^2 + 18x$$.
  15. Factor $$10y^2 – 20y$$.
  16. Factor $$2z^2 + 4z$$.
  17. Factor $$8x^2 + 24x$$.
  18. Factor $$15m^2 + 30m$$.
  19. Factor $$12n^2 – 36n$$.
  20. Factor $$14k^2 + 28k$$.

Hard Level Questions

  1. Factor $$x^2 + 5x + 6$$.
  2. Factor $$x^2 – 9$$.
  3. Factor $$x^2 – 5x + 6$$.
  4. Factor $$x^2 + 4x + 4$$.
  5. Factor $$x^2 + 6x + 8$$.
  6. Factor $$2x^2 – 4x – 6$$.
  7. Factor $$3x^2 + 12x + 12$$.
  8. Factor $$4x^2 – 16$$.
  9. Factor $$x^2 – 4x – 12$$.
  10. Factor $$2x^2 + 8x + 6$$.
  11. Factor $$5x^2 + 15x – 10$$.
  12. Factor $$4x^2 + 12x + 9$$.
  13. Factor $$6x^2 – 9x$$.
  14. Factor $$x^2 + 7x + 10$$.
  15. Factor $$x^2 + 2x – 15$$.
  16. Factor $$x^2 – 6x + 8$$.
  17. Factor $$3x^2 – 18$$.
  18. Factor $$x^2 – 2x – 15$$.
  19. Factor $$2x^2 + 4x – 6$$.
  20. Factor $$x^2 + 3x – 10$$.

Answers

Easy Level Answers

  1. $$3(x + 2)$$.
  2. $$8(y + 2)$$.
  3. $$5(a + 2)$$.
  4. $$4(x – 2)$$.
  5. $$12(y + 2)$$.
  6. $$10(x + 2)$$.
  7. $$6(x – 2)$$.
  8. $$9(a + 2)$$.
  9. $$2(x + 5)$$.
  10. $$7(b + 2)$$.
  11. $$15(m + 2)$$.
  12. $$20(y – 2)$$.
  13. $$18(x + 2)$$.
  14. $$11(n + 2)$$.
  15. $$13(p – 3)$$.
  16. $$6(k + 2)$$.
  17. $$2(x – 2)$$.
  18. $$4(y + 2)$$.
  19. $$5(x + 3)$$.
  20. $$2(a + 2)$$.

Medium Level Answers

  1. $$x(x + 2)$$.
  2. $$3x(x + 2)$$.
  3. $$2x(x – 2)$$.
  4. $$4x(x + 2)$$.
  5. $$5x(x – 2)$$.
  6. $$6y(y + 2)$$.
  7. $$8m(m + 2)$$.
  8. $$9n(n – 3)$$.
  9. $$7k(k + 2)$$.
  10. $$2x(x + 5)$$.
  11. $$4a(a – 2)$$.
  12. $$3b(b + 4)$$.
  13. $$5p(p – 3)$$.
  14. $$6x(x + 3)$$.
  15. $$10y(y – 2)$$.
  16. $$2z(z + 2)$$.
  17. $$8x(x + 3)$$.
  18. $$15m(m – 2)$$.
  19. $$12n(n – 3)$$.
  20. $$14k(k + 2)$$.

Hard Level Answers

  1. $$(x + 2)(x + 3)$$.
  2. $$(x – 3)(x + 3)$$.
  3. $$(x – 2)(x – 3)$$.
  4. $$(x + 2)(x + 2)$$.
  5. $$(x + 2)(x + 4)$$.
  6. $$2(x – 3)(x + 1)$$.
  7. $$3(x + 2)(x + 2)$$.
  8. $$(2x – 4)(2x + 4)$$.
  9. $$(x – 6)(x + 2)$$.
  10. $$2(x + 3)(x + 1)$$.
  11. $$5(x + 3)(x – 2)$$.
  12. $$(2x + 3)(2x + 3)$$.
  13. $$3(x – 3)(x + 1)$$.
  14. $$(x – 5)(x + 2)$$.
  15. $$(x – 5)(x + 3)$$.
  16. $$(x – 5)(x + 3)$$.
  17. $$3(x – 3)(x + 3)$$.
  18. $$(x – 5)(x + 3)$$.
  19. $$2(x + 3)(x – 1)$$.
  20. $$(x + 5)(x – 2)$$.

Happy learning, Year 7! Keep practicing, and you’ll master the factors of linear expressions in no time!