Find all the roots: the Rational Root Theorem, conjugate pairs and Descartes' Rule of Signs

Answer Key - Grade 11 Math Worksheet

Circle the best answer, or write your answer on the line.

1. 1.

   Zee's robot tests candidates for 3x³ + 4x² − 13x + 6 = 0. One root is a fraction. What is it?

   2/3
2. 2.

   Esme tests candidates for 3x³ − 4x² + 6x − 8 = 0 in a notebook. One root is a fraction. What is it?

   4/3
3. 3.

   Jun factors x⁴ + 7x² − 18 for math club. How many of its roots are real, and how many are complex (not real)?

   • two real roots and one complex root
   • four real roots and no complex roots
   • two real roots and no complex roots
   • two real roots and two complex roots
4. 4.

   Kira's robot factors x⁴ − 3x² − 4. How many of its roots are real, and how many are complex (not real)?

   • two real roots and two complex roots
   • two real roots and no complex roots
   • four real roots and no complex roots
   • two real roots and one complex root
5. 5.

   Luz factors x⁴ + 13x² + 36 on graph paper. How many of its roots are real, and how many are complex (not real)?

   • no real roots and two complex roots
   • no real roots and four complex roots
   • four real roots and no complex roots
   • no roots at all
6. 6.

   Milo factors x³ − 2x² + 9x − 18 by grouping. How many of its roots are real, and how many are complex (not real)?

   • one real root and no complex roots
   • one real root and one complex root
   • three real roots and no complex roots
   • one real root and two complex roots
7. 7.

   Noah uses Descartes' Rule of Signs on x⁴ − 3x³ + 2x − 5. How many positive real roots can it have?

   • three, two, one or none
   • two or none
   • three or one
   • three only
8. 8.

   Ren's robot uses Descartes' Rule of Signs on x⁵ − 2x⁴ − x³ + 4x² − 3x + 6. How many positive real roots can it have?

   • four, two or none
   • four, three, two, one or none
   • three or one
   • four only
9. 9.

   Tala uses Descartes' Rule of Signs on x³ + 2x² − 5x − 6. How many negative real roots can it have?

   • two only
   • one only
   • two, one or none
   • two or none
10. 10.

    Zee uses Descartes' Rule of Signs on 2x⁴ − x³ + 3x² + x − 4. How many negative real roots can it have?

    • one only
    • one or none
    • three or one