Polynomials quiz

The degree of the polynomial 3x^3 - 5x^2 + x - 7 is:

The zeroes of x^2 - 2x - 8 are:

The sum of the zeroes of 3x^2 - x - 4 is:

The product of the zeroes of 2x^2 + 7x - 4 is:

A quadratic polynomial whose zeroes are 3 and -2 is:

If the graph of y = p(x) cuts the x-axis at exactly two points, the number of zeroes of p(x) is:

The zeroes of 4s^2 - 4s + 1 are:

The zero of the linear polynomial 3x + 6 is:

The graph of a quadratic polynomial ax^2 + bx + c with a < 0 is a parabola that:

If one zero of x^2 - 7x + k is 3, the value of k is:

Find the zeroes of x^2 + 7x + 10 and verify the relationship between the zeroes and the coefficients. (2 marks)

Find a quadratic polynomial whose sum and product of zeroes are sqrt(2) and 1/3 respectively. (2 marks)

Find the zeroes of 3x^2 - x - 4 by factorisation. (2 marks)

Find a quadratic polynomial whose zeroes are -3 and 4. (2 marks)

If the zeroes of 2x^2 - 8x + k are equal, find k and the zeroes. (2 marks)

If alpha and beta are zeroes of x^2 - 6x + 8, find alpha^2 + beta^2. (2 marks)

Why is the expression x^2 + 1/x not a polynomial? Give a reason. (2 marks)

Find the quadratic polynomial whose zeroes are 0 and sqrt(5). (2 marks)

If one zero of 5x^2 + 13x + k is the reciprocal of the other, find k. (2 marks)

Can x - 1 be a factor of x^2 + x + 1? Justify using the idea of a zero. (2 marks)

Find the zeroes of the polynomial 2x^2 - 8x + 6 and verify the relationship between the zeroes and the coefficients. Then find the polynomial whose zeroes are double these zeroes. (3 marks)

The sum of the zeroes of a quadratic polynomial is 5 and the product is 6. Find the polynomial, its zeroes, and the value of the polynomial at x = 0. (3 marks)

If alpha and beta are the zeroes of 2x^2 - 5x - 3, find (i) alpha + beta (ii) alpha beta (iii) alpha^2 beta + alpha beta^2. (3 marks)

Find the zeroes of x^2 - (sqrt(3) + 1)x + sqrt(3) and verify the relationship between the zeroes and the coefficients. (3 marks)

If the zeroes of x^2 + px + q are 2 and -5, find p and q. Hence find the value of the polynomial at x = 1. (3 marks)

Read the passage and answer the questions. The arch of a footbridge over a canal is shaped like a parabola. Taking the water surface as the x-axis, the arch is described by y = -x^2 + 8x - 12, where x and y are in metres. (i) Find the points where the arch meets the water surface. (ii) What is the width of the arch at the water level? (iii) Is the parabola opening upward or downward? Give a reason from the polynomial. (5 marks)

Read the passage and answer the questions. A ball is thrown so that its height h (in m) after t seconds is given by h = -5t^2 + 20t. (i) Find the zeroes of the polynomial -5t^2 + 20t. (ii) What do the two zeroes mean physically? (iii) Find the sum and product of the zeroes using the coefficients and verify with (i). (5 marks)

Read the passage and answer the questions. A rectangular garden has an area of x^2 + 5x + 6 square metres, where x is a positive number. (i) Factorise the polynomial. (ii) Write possible expressions for the length and breadth of the garden. (iii) Find the zeroes of the polynomial and explain why they cannot be used directly as the sides. (5 marks)

Read the passage and answer the questions. A teacher writes the quadratic polynomial p(x) = x^2 - 4x + k on the board and says that its two zeroes differ by 2. (i) Write the sum of the zeroes. (ii) Using the given difference, find the two zeroes. (iii) Find the value of k. (5 marks)

Read the passage and answer the questions. In a mathematics exhibition, a student displays graphs of y = x^2 - 4, y = x^2 and y = x^2 + 4 on the same axes. (i) How many zeroes does each polynomial have? (ii) Find the zeroes of x^2 - 4. (iii) Explain, using the graphs, why x^2 + 4 has no real zero. (5 marks)

Draw the graph of p(x) = x^2 - x - 6 for x = -3 to 4. Read the zeroes from the graph, verify them by factorisation, and verify the relationship between the zeroes and the coefficients. State the coordinates of the lowest point of the graph. (6 marks)

Derive the relationship between the zeroes and the coefficients of a quadratic polynomial ax^2 + bx + c by writing it as k(x - alpha)(x - beta). Use it to find a quadratic polynomial whose zeroes are 2 + sqrt(3) and 2 - sqrt(3). (6 marks)

Verify that 4, -2 and 1/2 are the zeroes of the cubic polynomial 2x^3 - 5x^2 - 14x + 8. Then verify that the sum of zeroes is -b/a, the sum of the products of zeroes taken two at a time is c/a and the product of zeroes is -d/a. Sketch a rough graph showing the three zeroes. (6 marks)

Draw the graph of y = x^2 - 2x + 1 and y = x^2 - 2x + 3 on the same axes for x = -1 to 3. Compare the number of zeroes of the two polynomials and explain the difference using the graphs. (6 marks)

If alpha and beta are the zeroes of x^2 - 7x + 10, find the quadratic polynomial whose zeroes are 2 alpha and 2 beta, and the quadratic polynomial whose zeroes are 1/alpha and 1/beta. Verify your answers by finding the actual zeroes. (6 marks)