Polynomials

This chapter studies polynomials in one variable, algebraic expressions such as 2x^3 - 5x^2 + x - 7 in which the powers of the variable are whole numbers. You will learn the terms, coefficients and degree of a polynomial, and classify polynomials by the number of terms as monomials, binomials and trinomials, and by degree as constant, linear, quadratic and cubic. You will find the value of a polynomial p(x) at a given point and learn that a zero of a polynomial is a number r for which p(r) = 0; for example, 2 is a zero of x - 2. You will learn the Factor Theorem, which states that x - r is a factor of p(x) if p(r) = 0, and use it to factorise cubic polynomials. You will factorise quadratic polynomials by splitting the middle term and use algebraic identities, including the square of a trinomial, the cube of a binomial, and the identity for x^3 + y^3 + z^3 - 3xyz, to expand, evaluate and factorise expressions.

Detailed Worksheet: Polynomials Section A - Definitions (10 marks) 1. What is a polynomial in one variable? Is 4x^2 + 3/x a polynomial? Give a reason. (2 marks) 2. Define the degree of a polynomial. Write the degree of 5x^3 + 4x^2 + 7x and of 12. (2 marks) 3. Give one example each of a monomial, a binomial and a trinomial of degree 5 or less. (2 marks) 4. What is a zero of a polynomial? Find the zero of p(x) = 3x - 6. (2 marks) 5. State the Factor Theorem. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find p(0), p(1) and p(2) for p(x) = x^3 - 6x^2 + 11x - 6. What do you conclude about the factors of p(x)? (3 marks) 7. Factorise by splitting the middle term: (i) 2x^2 + 7x + 3 (ii) 6x^2 + 5x - 6. (3 marks) 8. Use the Factor Theorem to factorise x^3 - 23x^2 + 142x - 120. (3 marks) 9. Using suitable identities, evaluate: (i) 104 x 96 (ii) 103 x 107 (iii) 99^3. (3 marks) 10. Find the value of k if x - 1 is a factor of p(x) = kx^2 - 3x + k. (3 marks) Section C - Diagrams (10 marks) 11. Draw a square of side (x + y + z) divided into smaller squares and rectangles, and use it to show that (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx. (4 marks) 12. Draw a figure of a square of side (x + 3) divided into four parts and use it to show (x + 3)^2 = x^2 + 6x + 9. (3 marks) 13. The area of a rectangle is 25x^2 - 35x + 12. Factorise to find possible expressions for its length and breadth and draw the labelled rectangle. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Without actually calculating the cubes, find the value of (i) (-12)^3 + 7^3 + 5^3 (ii) 28^3 + (-15)^3 + (-13)^3, stating the identity used. (5 marks) 15. Factorise: (i) 27x^3 + y^3 + z^3 - 9xyz (ii) 8x^3 + 27y^3 + 36x^2y + 54xy^2. (5 marks) 16. The volume of a cuboid is 3x^2 - 12x. Find possible expressions for its dimensions. Also, if x + y + z = 0, show that x^3 + y^3 + z^3 = 3xyz. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. State the identity or theorem used in every question.