Algebraic Expressions

An algebraic expression is formed from variables and constants using operations such as addition, subtraction, multiplication and division, for example 3x + 5, 4xy - 7 or x^2 - 2x + 1. In this chapter you will learn that expressions are made up of terms, and each term is a product of factors. The numerical factor of a term is its coefficient, so in -5x^2y the coefficient is -5. Terms with the same algebraic factors, like 3xy and -7xy, are like terms, while 2x and 2y are unlike. Expressions with one, two or three unlike terms are monomials, binomials and trinomials, and in general polynomials. You will add and subtract expressions by combining like terms, as in (3a + 2b) + (5a - 4b) = 8a - 2b, and find the value of an expression by substituting values for the variables. Finally, you will use expressions to write formulas and rules, such as 2n + 1 for the nth odd number.

Detailed Worksheet: Algebraic Expressions Section A - Definitions (10 marks) 1. What is an algebraic expression? Give two examples. (2 marks) 2. Define a term and a factor. Write the terms and factors of 4x^2 - 3xy. (2 marks) 3. What is the coefficient of a term? Write the coefficient of x in (i) 5xy (ii) -x. (2 marks) 4. Distinguish between like and unlike terms with examples. (2 marks) 5. Define monomial, binomial and trinomial with one example of each. (2 marks) Section B - Calculations and Applications (15 marks) 6. Add: (i) 3mn, -5mn, 8mn and -4mn (ii) 7xy + 5yz - 3zx, 4yz + 9zx - 4y and -3xz + 5x - 2xy (iii) 5a^2 - 3ab + 2b^2 and -2a^2 + ab - 4b^2. (3 marks) 7. Subtract: (i) -5y^2 from y^2 (ii) 6xy from -12xy (iii) (a - b) from (a + b) (iv) 4a - 7ab + 3b + 12 from 12a - 9ab + 5b - 3. (3 marks) 8. If m = 2, find the value of (i) m - 2 (ii) 3m - 5 (iii) 9 - 5m (iv) 3m^2 - 2m - 7 (v) 5m/2 - 4. (3 marks) 9. If x = 2 and y = -1, find the value of (i) 2x + 2y (ii) x^2 - y^2 (iii) 2x^2 + y^2 - 3xy. (3 marks) 10. What should be added to x^2 + xy + y^2 to obtain 2x^2 + 3xy? What should be taken away from 3x^2 - 4y^2 + 5xy + 20 to obtain -x^2 - y^2 + 6xy + 20? (3 marks) Section C - Diagrams (10 marks) 11. Draw the first three figures of a pattern of squares made of matchsticks joined in a row (4, 7 and 10 matchsticks). Write an expression for the number of matchsticks in the nth figure and find it for n = 25. Draw a tree diagram showing the terms and factors of 3n + 1. (4 marks) 12. Draw a tree diagram showing the terms and factors of the expression 5xy^2 - 3x + 7. (3 marks) 13. Draw a rectangle with length (2x + 3) units and breadth (x - 1) units. Write an expression for its perimeter and simplify it. Find the perimeter when x = 4. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. A student simplified (4x - 3y) - (2x - 5y) as 2x - 8y. (i) Find the correct answer. (ii) Explain the mistake. (iii) State the rule for removing brackets preceded by a minus sign. (iv) Find the value of the correct answer when x = 3 and y = 2. (5 marks) 15. Look at the number pattern 3, 7, 11, 15, .... (i) Find the general rule (nth term) as an expression. (ii) Find the 20th term. (iii) Which term is 99? (iv) Explain why the rule has the form 4n plus a constant. (5 marks) 16. The sides of a triangle are (3x - 2y), (2x + y) and (x + 4y). (i) Find its perimeter. (ii) Find the perimeter when x = 5 and y = 2. (iii) Classify the perimeter expression as monomial, binomial or trinomial. (iv) If the perimeter is 6x + 3y + 10 after adding a fourth side to make a quadrilateral, find the fourth side. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Write x^2 to mean x squared, collect like terms carefully and show the substitution step in every evaluation.