2.11 Use of the factor theorem for rational values of the variable for polynomials Quiz: AQA Further Maths, Unit 2

The 20 questions

1. If f(2) = 0 for a polynomial f(x), which expression must be a factor of f(x)?

   • 2x
   • x + 2
   • x − 2
   • x

2. Which expression is a factor of x^3 − 4x?

   • x + 1
   • x + 2
   • x − 4
   • x^2 + 4

3. For f(x) = x^3 − 2x^2 + x, what is f(1)?

   • 2
   • 0
   • −1
   • 1

4. Which value of x shows that (2x + 1) is a factor of 2x^3 + 3x^2 − 3x − 2?

   • 1/2
   • −1
   • 2
   • −1/2

5. What value of k makes (x − 2) a factor of x^3 + kx − 6?

   • 1
   • −1
   • −2
   • 2

6. For which value of k is (x + 1) a factor of x^3 − 3x^2 + kx + 4?

   • 2
   • 0
   • 4
   • −4

7. Which is the full factorisation of x^3 + 2x^2 − 5x − 6?

   • (x + 1)(x + 3)(x − 2)
   • (x + 1)(x + 3)(x + 2)
   • (x + 1)(x − 3)(x + 2)
   • (x − 1)(x + 3)(x + 2)

8. What is the remainder when x^3 − 4x^2 + 2 is divided by (x − 3)?

   • −7
   • −1
   • 7
   • 2

9. Which is the full factorisation of 2x^3 − 5x^2 − 4x + 3?

   • (x − 3)(2x + 1)(x + 1)
   • (x − 3)(2x − 1)(x + 1)
   • (x − 3)(2x + 1)(x − 1)
   • (x + 3)(2x − 1)(x + 1)

10. Which value of k makes (x + 2) a factor of x^3 + kx^2 + 4?

    • 2
    • −1
    • 1
    • 3

11. Which factorisation is correct for 3x^3 − 4x^2 − 5x + 2?

    • (2x − 1)(x − 2)(x + 1)
    • (3x − 1)(x − 2)(x + 1)
    • (3x + 1)(x − 2)(x + 1)
    • (x + 3)(x − 2)(3x + 1)

12. For f(x) = x^3 + x^2 − 2x + 5, what is the remainder when f(x) is divided by (x + 2)?

    • 13
    • 5
    • 1
    • −5

13. Which set contains all the solutions of x^3 − 7x + 6 = 0?

    • {1, 2, −3}
    • {1, 2, 3}
    • {1, −2, 3}
    • {−1, 2, 3}

14. For what value of a is x^3 + ax − 6 divisible by (x − 3)?

    • 3
    • 7
    • −7
    • −3

15. The polynomial x^3 + px^2 − x + q has factors (x − 1) and (x + 2). What is p?

    • 3
    • 2
    • 1
    • −2

16. Which expression is a full factorisation of x^3 − 6x^2 + 11x − 6?

    • (x − 1)(x − 2)(x + 3)
    • (x − 1)(x + 2)(x − 3)
    • (x − 1)(x − 2)(x − 3)
    • (x + 1)(x − 2)(x − 3)

17. What value of m makes 2x^3 − mx^2 + 5 have a remainder of 0 when divided by (x − 1)?

    • 3
    • 7
    • −7
    • −3

18. What is the remainder when 2x^3 − x^2 + 3x − 4 is divided by (2x + 1)?

    • −6
    • −4
    • 6
    • 2

19. How many distinct real roots does x^3 − x^2 − 4x + 4 = 0 have?

    • 4
    • 1
    • 3
    • 2

20. Which expression is a factor of x^3 − 7x + 6?

    • x + 7
    • x − 6
    • x − 2
    • x + 2

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