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

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