Factorisation

Factorisation is the reverse of multiplication: we write an algebraic expression as a product of its factors. Just as 30 = 2 x 3 x 5, the term 12xy can be written as 2 x 2 x 3 x x x y, its irreducible factors. In this chapter you will factorise expressions by taking out common factors, as in 10x^2y - 15xy^2 = 5xy(2x - 3y), and by regrouping terms so that a common binomial factor appears, as in xy + 2x + 3y + 6 = (x + 3)(y + 2). You will also factorise using the identities for perfect squares and for the difference of two squares, such as 49p^2 - 36 = (7p - 6)(7p + 6), and expressions of the form x^2 + (a + b)x + ab by finding two numbers with a given sum and product. Finally, you will divide a monomial by a monomial, a polynomial by a monomial and a polynomial by a polynomial after factorising, and learn to find and correct common errors in algebra.

Detailed Worksheet: Factorisation Section A - Definitions (10 marks) 1. What is meant by factorisation of an algebraic expression? Write the irreducible factors of 12xy. (2 marks) 2. Explain the common factor method of factorisation with the example 12x + 36. (2 marks) 3. What is factorisation by regrouping terms? When is it used? Give one example. (2 marks) 4. Write the three algebraic identities that are commonly used for factorisation. (2 marks) 5. Why is division of algebraic expressions called the inverse of multiplication? Explain with 6x^2 / 2x. (2 marks) Section B - Calculations and Applications (15 marks) 6. Factorise by taking out common factors: (i) 6p - 12q (ii) 7m^2 + 14m (iii) 10x^2y - 15xy^2 + 20xy. (3 marks) 7. Factorise by regrouping: (i) xy + 2x + 3y + 6 (ii) 15pq + 15 + 9q + 25p (iii) z - 7 + 7xy - xyz. (3 marks) 8. Factorise using identities: (i) 4y^2 - 12y + 9 (ii) 49p^2 - 36 (iii) x^4 - 256. (3 marks) 9. Factorise: (i) p^2 + 6p + 8 (ii) q^2 - 10q + 21 (iii) z^2 - 4z - 12. (3 marks) 10. Carry out the divisions: (i) 28x^4 / 56x (ii) (5x^2 - 6x) / 3x (iii) (y^2 + 7y + 10) / (y + 5). (3 marks) Section C - Diagrams (10 marks) 11. Using algebra tiles (one x by x square, strips of x by 1 and unit squares), draw how the tiles for x^2 + 5x + 6 can be arranged into a rectangle. Label its sides and write the factorisation. (4 marks) 12. Draw a square of side x from which a square of side 3 has been removed from one corner. Show by cutting and rearranging that x^2 - 9 = (x + 3)(x - 3). (3 marks) 13. Draw factor trees showing the irreducible factors of 30x^2y and 42xy^2, and use them to find the common factors of the two terms. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Find and correct the errors: (i) 4(x - 5) = 4x - 5 (ii) x(3x + 2) = 3x^2 + 2 (iii) (3x + 2)/3x = 2 (iv) (2x)^2 + 4(2x) + 7 = 2x^2 + 8x + 7 (v) (y - 3)^2 = y^2 - 9. (5 marks) 15. The area of a rectangle is (x^2 + 7x + 12) cm^2. (i) Factorise to find possible expressions for its length and breadth. (ii) Find an expression for its perimeter. (iii) Find the length, breadth and area when x = 5, and check that the area agrees with the original expression. (5 marks) 16. Divide by first factorising the expressions: (i) 39y^3(50y^2 - 98) by 26y^2(5y + 7) (ii) (m^2 - 14m - 32) by (m + 2) (iii) 44(x^4 - 5x^3 - 24x^2) by 11x(x - 8). (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Check every factorisation by multiplying the factors back, and show each step of division.