Lesson 6: Theorems About Roots of Polynomial Equations — Practice Questions

1. 1. A polynomial equation with rational coefficients has a root of $3 + \sqrt{7}$. Which of the following must also be a root of the equation?

   • A. $3 - \sqrt{7}$
   • B. $-3 + \sqrt{7}$
   • C. $\sqrt{7} - 3$
   • D. $3 + \sqrt{-7}$

2. 2. A polynomial with rational coefficients has a root of $4 - \sqrt{11}$. According to the Irrational Conjugate Root Theorem, another root must be ___.

3. 3. If a polynomial with rational coefficients has a root of $-2 + 5\sqrt{3}$, what is the conjugate root that must also exist?

4. 4. The Irrational Conjugate Root Theorem requires a specific condition for the polynomial. What is this condition?

   • A. The polynomial must have integer coefficients.
   • B. The polynomial must have rational coefficients.
   • C. The polynomial must be of an even degree.
   • D. The polynomial must have a positive leading coefficient.

5. 5. Two roots of a polynomial with rational coefficients are $6 + \sqrt{5}$ and $6 - \sqrt{5}$. What is the product of these two conjugate roots? ___

6. 6. A polynomial has rational coefficients. If one of its roots is $4 - \sqrt{7}$, which of the following must also be a root of the polynomial?

   • A. $4 + \sqrt{7}$
   • B. $-4 + \sqrt{7}$
   • C. $1/(4 - \sqrt{7})$
   • D. $\sqrt{7} - 4$

7. 7. A polynomial with real coefficients has a root of $5 + 2i$. Which of the following must also be a root?

   • A. $5 - 2i$
   • B. $-5 + 2i$
   • C. $-5 - 2i$
   • D. $2 - 5i$

8. 8. What is the polynomial of lowest degree with roots at $x = -2$ and $x = 5$? Write your answer in expanded form $x^2 + Bx + C$. The polynomial is ___.

9. 9. A polynomial with rational coefficients has roots $2 + \sqrt{3}$ and $2 - \sqrt{3}$. The quadratic factor corresponding to these roots is $x^2 - 4x +$ ___.

10. 10. A polynomial with real coefficients has a root of $1 - 4i$. What is the quadratic factor that corresponds to this root and its conjugate? The factor is ___.