Polynomials

A polynomial in x is an algebraic expression such as 2x^2 - 3x + 5 in which the powers of x are whole numbers. The highest power is its degree, so polynomials of degree 1, 2 and 3 are called linear, quadratic and cubic. A real number k is a zero of p(x) if p(k) = 0. Geometrically, the zeroes are the x-coordinates of the points where the graph of y = p(x) meets the x-axis. A linear graph is a straight line with one zero, while a quadratic graph is a parabola that may cut the x-axis at two points, touch it at one point, or not meet it at all. You will then discover the relationship between the zeroes and coefficients of a quadratic polynomial ax^2 + bx + c: the sum of zeroes is -b/a and the product is c/a. Using these results you will verify zeroes found by factorisation, find a quadratic polynomial when the sum and product of its zeroes are given, and evaluate expressions built from the zeroes.

Detailed Worksheet: Polynomials Section A - Definitions (10 marks) 1. Define a polynomial and its degree. Give one example each of a linear, a quadratic and a cubic polynomial. (2 marks) 2. What is meant by a zero of a polynomial p(x)? Is 2 a zero of x^2 - 4x + 4? Justify. (2 marks) 3. What is the geometrical meaning of the zeroes of a polynomial p(x)? (2 marks) 4. Write the relationship between the zeroes and the coefficients of the quadratic polynomial ax^2 + bx + c. (2 marks) 5. What is the maximum number of zeroes a polynomial of degree n can have? Explain with reference to linear and quadratic graphs. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the zeroes of 6x^2 - 3 - 7x and verify the relationship between the zeroes and the coefficients. (3 marks) 7. Find the zeroes of 4u^2 + 8u and of t^2 - 15, and verify the relationship between the zeroes and the coefficients in each case. (3 marks) 8. Find a quadratic polynomial whose sum and product of zeroes are 1/4 and -1 respectively. Also check whether x = 1 is a zero of this polynomial. (3 marks) 9. If alpha and beta are the zeroes of x^2 - 5x + 6, find the values of (i) alpha^2 + beta^2 (ii) 1/alpha + 1/beta (iii) alpha - beta, given alpha > beta. (3 marks) 10. If -3 is one zero of x^2 + kx - 6, find k and the other zero. (3 marks) Section C - Diagrams (10 marks) 11. Draw the graph of y = x^2 - 3x - 4 for x from -2 to 5. From the graph, read off the zeroes of the polynomial and verify them by factorisation. (4 marks) 12. Draw the graph of y = 2x + 3 and mark the point where it cuts the x-axis. Explain why a linear polynomial has exactly one zero. (3 marks) 13. Sketch rough graphs of three quadratic polynomials with a > 0 that have (i) two distinct zeroes (ii) one repeated zero (iii) no zero. State the number of zeroes in each case. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. The graph of y = p(x) is given for four polynomials: (i) a line parallel to the x-axis, (ii) a parabola cutting the x-axis at two points, (iii) a curve cutting the x-axis at three points, (iv) a parabola opening downward that does not meet the x-axis. State the number of zeroes in each case and what the degree could be. Explain why a quadratic polynomial cannot have three zeroes. (5 marks) 15. For what value of k is the sum of the zeroes of kx^2 + 2x + 3k equal to their product? A student says k = 2/3. Check the student's answer and correct it if wrong. (5 marks) 16. Find a quadratic polynomial whose zeroes are 3 + sqrt(2) and 3 - sqrt(2). Show that its coefficients are integers even though the zeroes are irrational, and explain why. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show every step of factorisation. Draw graphs on squared paper with a suitable scale and label the axes.