Year 8 Maths: Factorising

What is Factorising?

Factorising is a process in maths where we take an expression and write it as a product of its factors. Factors are numbers or expressions that can be multiplied together to get another number or expression. For example, if we have the expression ( x^2 + 5x ), we can factor it into ( x(x + 5) ).

Why do we Factorise?

Factorising is useful because it helps us simplify expressions, solve equations, and make it easier to work with algebra. It helps us understand the relationships between numbers and variables better. By factorising, we can also find the roots of equations, which are the values that make the equation equal to zero.

Key Rules for Factorising

1. Look for Common Factors: Always check if there is a number or variable that appears in all parts of the expression.Example: In ( 4x + 8 ), both terms have a common factor of 4.So, we can factor it as ( 4(x + 2) ).
2. Difference of Squares: If you have an expression like ( a^2 – b^2 ), it can be factored as ( (a – b)(a + b) ).Example: ( x^2 – 9 ) can be factored into ( (x – 3)(x + 3) ).
3. Quadratic Expressions: For a quadratic expression like ( ax^2 + bx + c ), we look for two numbers that multiply to ( ac ) and add to ( b ).Example: For ( x^2 + 5x + 6 ), we find ( 2 ) and ( 3 ), so we can factor it as ( (x + 2)(x + 3) ).

Tips and Tricks

• Use a Grid: When factorising quadratics, you can use a grid to help visualise how the factors combine.
• Practice Regularly: The more you practice factorising, the better you will get at spotting patterns and factors.
• Check Your Work: After factorising, you can always expand your factors back out to check if you get the original expression.

Questions

Easy Level Questions

1. Factorise ( 3x + 6 ).
2. Factorise ( 5y + 15 ).
3. Factorise ( 2a + 8 ).
4. Factorise ( x^2 + 4x ).
5. Factorise ( 2b^2 + 6b ).
6. Factorise ( 8 + 4x ).
7. Factorise ( 10 – 5y ).
8. Factorise ( 12z + 18 ).
9. Factorise ( 6x^2 + 9x ).
10. Factorise ( 14m + 21n ).
11. Factorise ( x^2 + 2x + 1 ).
12. Factorise ( x^2 – 4 ).
13. Factorise ( 3xy + 6x ).
14. Factorise ( 7a + 14b ).
15. Factorise ( 9p + 12q ).
16. Factorise ( 4x^2 + 8x ).
17. Factorise ( 15k + 25 ).
18. Factorise ( 16 – x^2 ).
19. Factorise ( 5x + 10y ).
20. Factorise ( 3a^2 + 6a ).

Medium Level Questions

1. Factorise ( x^2 + 6x + 9 ).
2. Factorise ( x^2 – 10x + 21 ).
3. Factorise ( 2x^2 + 8x ).
4. Factorise ( 4x^2 – 16 ).
5. Factorise ( 3x^2 + 12x + 12 ).
6. Factorise ( x^2 + 5x – 6 ).
7. Factorise ( 2x^2 + 3x – 5 ).
8. Factorise ( x^2 – 9x + 20 ).
9. Factorise ( x^2 + 3x – 10 ).
10. Factorise ( 6x^2 + 11x – 10 ).
11. Factorise ( 7x^2 – 14x ).
12. Factorise ( 5x^2 + 15x ).
13. Factorise ( x^2 + 2x – 8 ).
14. Factorise ( 10x^2 – 25 ).
15. Factorise ( x^2 – 5x – 14 ).
16. Factorise ( 4y^2 – 25 ).
17. Factorise ( 9a^2 – 1 ).
18. Factorise ( 8x^2 + 4x – 12 ).
19. Factorise ( 15m^2 – 45m ).
20. Factorise ( 2x^2 – 8x + 6 ).

Hard Level Questions

1. Factorise ( x^3 – 27 ).
2. Factorise ( 4x^2 – 12x + 9 ).
3. Factorise ( x^2 – 10x + 25 ).
4. Factorise ( 3x^2 + 6x – 9 ).
5. Factorise ( 2x^3 + 4x^2 – 6x ).
6. Factorise ( a^2 + 10a + 21 ).
7. Factorise ( x^4 – 16 ).
8. Factorise ( 2x^4 – 8x^2 ).
9. Factorise ( 5a^2 – 20a + 15 ).
10. Factorise ( 3x^2 – 12x + 12 ).
11. Factorise ( x^3 + 3x^2 – 4x – 12 ).
12. Factorise ( 8x^2 – 18x + 9 ).
13. Factorise ( x^2 + 2x + 1 – 9 ).
14. Factorise ( 12x^2 – 20x ).
15. Factorise ( 4x^4 – 16x^2 ).
16. Factorise ( 10x^2 + 15x – 25 ).
17. Factorise ( 2x^2 + 10x + 12 ).
18. Factorise ( 9x^2 + 30x + 25 ).
19. Factorise ( x^2 – 6x + 8 ).
20. Factorise ( 2x^3 + 8x^2 + 6x ).

Answers and Explanations

Easy Level Answers

1. ( 3(x + 2) ) — We find that 3 is a common factor.
2. ( 5(y + 3) ) — Here, both terms can be divided by 5.
3. ( 2(a + 4) ) — The number 2 is a factor of both terms.
4. ( x(x + 4) ) — The common factor is ( x ).
5. ( 2b(b + 3) ) — Factor 2 from both terms.
6. ( 4(2 + x) ) — Factor out the 4.
7. ( 5(2 – y) ) — Here, we can factor out 5.
8. ( 6(2z + 3) ) — The common factor is 6.
9. ( 3x(2x + 3) ) — Factor out 3x.
10. ( 7(m + 3n) ) — We can factor 7 out.
11. ( (x + 1)^2 ) — This is a perfect square.
12. ( (x – 2)(x + 2) ) — This is a difference of squares.
13. ( 3x(y + 2) ) — Here 3x is the common factor.
14. ( 7(a + 2b) ) — Factor out 7.
15. ( 3(3p + 4q) ) — The common factor is 3.
16. ( 4x(x + 2) ) — Factor out 4x.
17. ( 5(3k – 5) ) — Factor out 5.
18. ( (4 – x)(4 + x) ) — This is a difference of squares.
19. ( 5(x + 2y) ) — Factor out 5.
20. ( 3a(a + 2) ) — Factor out 3a.

Medium Level Answers

1. ( (x + 3)^2 ) — This is a perfect square trinomial.
2. ( (x – 3)(x – 7) ) — Two numbers that multiply to 21 and add to -10.
3. ( 2x(x + 4) ) — Factor out 2x.
4. ( 4(x – 2)(x + 2) ) — This is a difference of squares.
5. ( 3(x + 2)^2 ) — Recognise the perfect square.
6. ( (x – 1)(x + 6) ) — Find two numbers that multiply to -6 and add to 5.
7. ( (2x – 1)(x + 5) ) — Two numbers that multiply to -10.
8. ( (x – 4)(x – 5) ) — Find the roots of the equation.
9. ( (x + 5)(x – 2) ) — Two numbers that multiply to -10 and add to 3.
10. ( (3x – 2)(2x + 5) ) — Find numbers that multiply to -60.
11. ( 7x(x – 2) ) — Factor out 7x.
12. ( 5x(x + 3) ) — Factor out 5x.
13. ( (x + 4)(x – 2) ) — Two numbers that multiply to -8.
14. ( 5(2x – 5) ) — Factor out 5.
15. ( (x – 7)(x + 2) ) — Find the roots of the equation.
16. ( (2y – 5)(2y + 5) ) — This is a difference of squares.
17. ( (3a – 1)(3a + 1) ) — This is a difference of squares.
18. ( 2(x^2 – 9) ) — Recognise it as a difference of squares.
19. ( 3(5m – 15) ) — Factor out 3.
20. ( 2(x^2 – 4x + 3) ) — Factor out 2.

Hard Level Answers

1. ( (x – 3)(x^2 + 3x + 9) ) — This is a difference of cubes.
2. ( 2(x – 3)^2 ) — This is a perfect square.
3. ( (x – 5)^2 ) — Recognise it as a perfect square.
4. ( 3(x – 1)(x + 3) ) — Find the roots of the equation.
5. ( 2x(x^2 + 2x – 3) ) — Factor out 2x first.
6. ( (a + 3)(a + 7) ) — Two numbers that multiply to 21.
7. ( (x^2 – 4)(x^2 + 4) ) — Factor the difference of squares.
8. ( 2x^2(x^2 – 4) ) — Factor out ( 2x^2 ) first.
9. ( 5(a – 3)(a – 1) ) — Find the roots of the quadratic.
10. ( 3(x – 2)(x – 2) ) — This is a perfect square.
11. ( (x + 4)(x^2 – 3) ) — Find two factors.
12. ( (2x – 3)(4x – 3) ) — Find the roots of the quadratic.
13. ( (x + 3)(x – 3) ) — This is a difference of squares.
14. ( 4(3x – 5)(x + 1) ) — Factor out the 4 first.
15. ( 2(x – 5)(x + 5) ) — This is a difference of squares.
16. ( (5x + 15)(x – 2) ) — Factor out the 5.
17. ( 2(x^2 + 5x + 6) ) — Factor out 2 first.
18. ( (3x + 5)^2 ) — Recognise it as a perfect square.
19. ( (x – 4)(x – 2) ) — Find the roots of the quadratic.
20. ( 2x(x^2 + 4x + 3) ) — Factor out ( 2x ) first.

With this guide, you should now have a clearer understanding of factorising. Remember, practice makes perfect! Keep working through these problems, and you’ll become more confident in your factorising skills.