11th grade Find all the roots: the Rational Root Theorem, conjugate pairs and Descartes' Rule of Signs worksheet

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

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. 2.

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