Question 1

According to the n-Root Theorem, how many roots (including multiplicity) does the polynomial $P(x) = x^4 - 8x + 16$ have?

Accepted Answer

$4$

Question 2

Given the polynomial $f(x) = (x-1)(x-4)(x-4)$, what are the roots and their multiplicities?

Accepted Answer

Roots: $x=1$ (multiplicity 1), $x=4$ (multiplicity 2)

Question 3

Which of the following is a possible set of rational roots for $g(x) = x^3 - 3x^2 + 7x - 6$ according to the Rational Root Theorem?

Accepted Answer

$\pm1, \pm2, \pm3, \pm6$

Question 4

If a polynomial with real coefficients has a root $x = 2 + 3i$, which other root must it also have?

Accepted Answer

$x = 2 - 3i$

Question 5

What is the degree of the polynomial $f(x) = x^2(x^2 + 1)$?

Accepted Answer

$4$

Question 6

Given the quadratic equation $x^2 + 2x + 5 = 0$, what are its roots?

Accepted Answer

$x = -1 \pm 2i$

Question 7

A chemical reaction is modeled by the polynomial $x^3 - 4x^2 + 5x - 2 = 0$. If $x=1$ is a root, which factor must divide the polynomial?

Accepted Answer

$(x-1)$

Question 8

If the roots of a polynomial are $x = 0$, $x = 2i$, and $x = -2i$, what is the polynomial with real coefficients?

Accepted Answer

$x(x^2 + 4)$

Question 9

A medical dosage model uses the equation $x^2 - 2x + 5 = 0$. What is the sum of the roots?

Accepted Answer

$2$

Question 10

Which of the following polynomials has $x = 3$ as a double root?

Accepted Answer

$(x-3)^2(x+1)$

Question 11

If $x = 1 + 2i$ is a root of a polynomial with real coefficients, which of the following must also be a root?

Accepted Answer

$x = 1 - 2i$

Question 12

Find the polynomial of least degree with real coefficients that has roots $x = 2$ and $x = 1 + i$.

Accepted Answer

$(x-2)(x^2 - 2x + 2)$

Question 13

A polynomial $f(x)$ has roots $x = 0$, $x = 2$, and $x = -2$. Which of the following is $f(x)$?

Accepted Answer

$x(x-2)(x+2)$

Question 14

If $x^2 + 4 = 0$, what are the roots?

Accepted Answer

$x = \pm 2i$

Question 15

Which of the following is the correct quadratic formula for solving $ax^2 + bx + c = 0$?

Accepted Answer

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Question 16

A polynomial $f(x)$ has roots $x = 3$ and $x = -3i$. What is the minimum degree of $f(x)$ with real coefficients?

Accepted Answer

$3$

Question 17

If $x = 2$ is a root of multiplicity 2 and $x = -1$ is a root of multiplicity 1, which polynomial matches these roots?

Accepted Answer

$(x-2)^2(x+1)$

Question 18

A polynomial $f(x)$ has roots $x = 1$, $x = 2$, and $x = 3$. Which of the following is $f(x)$?

Accepted Answer

$(x-1)(x-2)(x-3)$

Question 19

If a polynomial has roots $x = 1 + \sqrt{2}$ and $x = 1 - \sqrt{2}$, what is the quadratic polynomial with real coefficients?

Accepted Answer

$(x-1)^2 - 2$

Theory of Equations: Finding Roots and Writing Polynomial Equations

Study Guide - Practice Questions