Algebraic Expressions and Identities

Algebraic expressions are built from variables and constants using addition, subtraction and multiplication. In this chapter you will identify the terms of an expression such as 5x^2y - 3xy + 7, the factors of each term and its numerical coefficient, and classify expressions as monomials, binomials, trinomials or polynomials. Only like terms can be added or subtracted, so you will practise arranging like terms in columns. You will then multiply monomials, a monomial by a polynomial and a binomial by a binomial, and use these products to find areas of rectangles and volumes of cuboids. The second part introduces identities, equalities that hold for every value of the variables. You will learn (a + b)^2 = a^2 + 2ab + b^2, (a - b)^2 = a^2 - 2ab + b^2, (a + b)(a - b) = a^2 - b^2 and (x + a)(x + b) = x^2 + (a + b)x + ab, verify them geometrically and numerically, and use them for quick calculations such as 103^2 and 197 x 203.

Detailed Worksheet: Algebraic Expressions and Identities Section A - Definitions (10 marks) 1. Define a term, a factor and a numerical coefficient of an algebraic expression. Write the terms of 5xy^2 - 3x + 7 and the coefficient of each term. (2 marks) 2. Define a monomial, a binomial and a trinomial. Classify each of the following: (i) 4y - 7z (ii) x + y - xy (iii) 1000 (iv) pqr. (2 marks) 3. What are like terms and unlike terms? From the list 3xy, -4x^2y, 7yx, 2xy^2, -xy, pick out all the like terms. (2 marks) 4. What is an identity? How is an identity different from an equation? Give one example of each. (2 marks) 5. Write the four standard identities studied in this chapter. (2 marks) Section B - Calculations and Applications (15 marks) 6. Add 5x^2 - 3xy + 2y^2, -2x^2 + 7xy - 4y^2 and x^2 - xy + 6. Arrange like terms in columns. (3 marks) 7. Find the products: (i) (3x - 4y)(2x + 5y) (ii) (2p^2 - 3)(p + 4) (iii) 4mn(m^2 - 2mn + 3n^2). (3 marks) 8. Using suitable identities, evaluate: (i) 103^2 (ii) 98^2 (iii) 197 x 203. (3 marks) 9. Using the identity (x + a)(x + b) = x^2 + (a + b)x + ab, find: (i) 103 x 104 (ii) (y + 6)(y - 2) (iii) 5.1 x 5.2. (3 marks) 10. If x + y = 12 and xy = 32, find the values of (i) x^2 + y^2 (ii) (x - y)^2. (3 marks) Section C - Diagrams (10 marks) 11. Draw a square of side (p + q) and divide it into two squares and two rectangles. Label the area of each part and use the diagram to show that (p + q)^2 = p^2 + 2pq + q^2. (4 marks) 12. Draw a rectangle of length (x + 5) units and breadth (x + 3) units, divided into four parts. Label the area of each part and deduce that (x + 5)(x + 3) = x^2 + 8x + 15. (3 marks) 13. Draw a square of side p from which a smaller square of side q has been cut out of one corner. Show by cutting and rearranging the remaining piece that p^2 - q^2 = (p + q)(p - q). (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. A student wrote (x + 3)^2 = x^2 + 9. Check the statement by substituting x = 2, identify the error and write the correct expansion. In the same way, explain why (2x - 3y)^2 is not equal to 4x^2 - 9y^2 and expand it correctly. (5 marks) 15. A rectangular garden has length (3x + 2) m and breadth (2x - 1) m. (i) Find an expression for its area. (ii) Find an expression for its perimeter. (iii) Find the actual area and perimeter when x = 4, and verify your answers by first finding the length and breadth. (5 marks) 16. Prove the following: (i) (3m + 4n)^2 - (3m - 4n)^2 = 48mn (ii) (p - q)(p + q) + (q - r)(q + r) + (r - p)(r + p) = 0 (iii) (x + 2)(x + 5) - (x + 3)(x + 4) = -2 for every value of x. Explain why result (iii) is called an identity. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show every step of multiplication and simplification, and state the identity used wherever one is applied.