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

Answer Key - Grade 11 Math Worksheet - Challenge Me

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

1. 1.

   Bo finds that 2x³ − x² + 18x − 9 has one rational root. What are all its roots?

   1/2, 3i, -3i
2. 2.

   Esme's robot finds one rational root of x³ + 3x² + 4x + 12. What are all its roots?

   -3, 2i, -2i
3. 3.

   Jun finds that 3x³ + 2x² + 3x + 2 has one rational root. What are all its roots?

   -2/3, i, -i
4. 4.

   Kira finds one rational root of x³ − 4x² + 25x − 100. What are all its roots?

   4, 5i, -5i
5. 5.

   Milo's robot needs a degree 3 polynomial with real coefficients, leading coefficient 1, and roots -1 and 2 − i. What is it in standard form?

   x³ − 3x² + x + 5
6. 6.

   Noah needs a degree 3 polynomial with real coefficients, leading coefficient 1, and roots 3 and -2 + i. What is it in standard form?

   x³ + x² − 7x − 15
7. 7.

   Tala's polynomial has degree 6 and real coefficients. Two roots are 7i and 3 + 8i; every other root is real. How many real roots does it have, counting repeats?

   2
8. 8.

   Zee's robot has a degree 7 polynomial with real coefficients. Two roots are 6i and 4 − 8i; the rest are real. How many real roots, counting repeats?

   3
9. 9.

   Bo has a degree 5 polynomial with real coefficients. One root is 9i, and every other root is real. How many real roots does it have, counting repeats?

   3
10. 10.

    Esme has a degree 10 polynomial with real coefficients. Two roots are 4i and 9 + 2i; the rest are real. How many real roots, counting repeats?

    6