4.1-4.3 Roots of polynomials and forming polynomials with given roots Quiz: Pearson Edexcel Further Maths, Unit 4

The 20 questions

1. For the cubic x^3 + px^2 + qx + r = 0 with roots alpha, beta, gamma, what is alpha + beta + gamma?

   • p
   • -q
   • q
   • -p

2. For the cubic ax^3 + bx^2 + cx + d = 0, what is the product of its roots?

   • d/a
   • -b/a
   • -d/a
   • -c/a

3. For the quadratic ax^2 + bx + c = 0 with roots alpha and beta, what is alpha beta?

   • -c/a
   • c/a
   • b/a
   • -b/a

4. For the cubic x^3 + px^2 + qx + r = 0 with roots alpha, beta, gamma, what is alpha beta + beta gamma + gamma alpha?

   • -r
   • q
   • p
   • -q

5. How many roots, counting multiplicity, does a cubic polynomial have over the complex numbers?

   • 4
   • 1
   • 3
   • 2

6. A real cubic polynomial must have at least how many real roots?

   • 1
   • 0
   • 2
   • 3

7. For the cubic x^3 + px^2 + qx + r = 0 with roots alpha, beta, gamma, which expression gives alpha^2 + beta^2 + gamma^2?

   • p^2 - 2q
   • p^2 + 2q
   • q^2 - 2p
   • p^2 - q

8. What is alpha^2 + beta^2 + gamma^2 for the roots of x^3 - 6x^2 + 11x - 6 = 0?

   • 22
   • 11
   • 14
   • 36

9. What is alpha^2 + beta^2 + gamma^2 for the roots of x^3 - 2x + 5 = 0?

   • -4
   • 2
   • 10
   • 4

10. What is 1/alpha + 1/beta + 1/gamma for the roots of x^3 - 6x^2 + 11x - 6 = 0?

    • 6/11
    • -11/6
    • 1
    • 11/6

11. Which quadratic has roots equal to the squares of the roots of x^2 + 3x + 2 = 0?

    • x^2 - 3x + 4 = 0
    • x^2 + 5x + 4 = 0
    • x^2 - 5x + 4 = 0
    • x^2 - 5x + 2 = 0

12. Which quadratic has roots equal to the reciprocals of the roots of x^2 - 3x + 2 = 0?

    • x^2 - 2x + 3 = 0
    • 2x^2 - 3x + 1 = 0
    • 2x^2 + 3x + 1 = 0
    • x^2 - 3x + 2 = 0

13. Let alpha be a root of x^2 - 2x - 3 = 0. Which equation has roots alpha + 1?

    • y^2 - 4y = 0
    • y^2 + 4y = 0
    • y^2 - 2y - 3 = 0
    • y^2 - 4y + 4 = 0

14. The roots of x^2 - 5x + 6 = 0 are 2 and 3. Which quadratic has roots twice these values?

    • y^2 - 5y + 12 = 0
    • y^2 - 5y + 6 = 0
    • y^2 - 10y + 6 = 0
    • y^2 - 10y + 24 = 0

15. If alpha is a root of x^3 - 2x^2 + 3x - 4 = 0, which polynomial has 1/alpha as a root?

    • 4x^3 + 3x^2 + 2x + 1 = 0
    • 4x^3 - 3x^2 + 2x - 1 = 0
    • 4x^3 - 2x^2 + 3x - 1 = 0
    • x^3 - 3x^2 + 2x - 4 = 0

16. For the roots alpha, beta, gamma of x^3 - 2x^2 + 3x - 4 = 0, what is alpha^3 + beta^3 + gamma^3?

    • 2
    • -2
    • 10
    • 6

17. For the roots alpha, beta, gamma of x^3 - 2x^2 + 3x - 4 = 0, what is alpha^2 + beta^2 + gamma^2?

    • -2
    • 2
    • 10
    • 4

18. For the same cubic x^3 - 2x^2 + 3x - 4 = 0, what is (3 + alpha)(3 + beta)(3 + gamma)?

    • 40
    • 22
    • -58
    • 58

19. For the roots alpha and beta of x^2 - 3x + 1 = 0, what is alpha^3 + beta^3?

    • 18
    • 27
    • 6
    • 9

20. For the cubic x^3 - 3x + 1 = 0, which polynomial has roots alpha - 1?

    • y^3 - 3y^2 - 1 = 0
    • y^3 + 3y^2 + 1 = 0
    • y^3 + 3y^2 - 1 = 0
    • y^3 + 3y - 1 = 0

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